Surgery and Geometric Topology

Home , Proper map, Surgery theory

SURGERY AND GEOMETRIC TOPOLOGY

Andrew Ranicki and Masayuki Yamasaki Editors

Proceedings of a conference held at Josai University

Sakado September

Contents

Foreword v

Program of the conference vii

List of participants ix

Toshiyuki Akita Cohomology and Euler characteristics of Coxeter groups

Frank Connolly and Bogdan Vajiac Completions of StratiedEnds

Zbigniew Fiedorowicz and Yong jin Song The braid structure of mapping class groups

Bruce Hughes Control led topological equivalence of maps in the theory of stratied

spaces and approximate brations

Tsuyoshi Kato The asymptotic method in the Novikov conjecture

Tomohiro Kawakami Anoteonexponential ly Nash G manifolds and vector bund les

Masaharu Morimoto On xedpoint data of smooth actions on spheres

Eiji Ogasa Some results on knots and links in al l dimensions

Erik K Pedersen Control led algebra and topology

Frank Quinn Problems in lowdimensional topology

Andrew Ranicki slides on chain duality

Andrew Ranicki and Masayuki Yamasaki Control led Ltheory preliminary announce

ment iii

iv CONTENTS

John Ro e Notes on surgery and C algebras

Tatsuhiko Yagasaki Characterizations of innitedimensional manifold tuples and

their applications to homeomorphism groups

Foreword

As the second of a conference series in mathematics Josai University sp onsored

the Conference on Surgery and Geometric Top ology during the week of September

The scientic program consisted of lectures listed b elow and there

was an excursion to Takaragawa Gunma on the st

This volume collects pap ers by participantsaswell as some of the abstracts pre

pared by the lecturers for the conference The articles are also available electronically

on WWW from

httpmathjosaiacjpyamasakiconferencehtml

at least for several years

Wewould like to thank Josai University and GrantinAid for Scientic Research

A of the Ministry of Education Science Sp orts and Culture of Japan for their

generous nancial supp ort Thanks are also due to the university sta for their various

supp ort to the lecturers and to the participants of the conference

Andrew Ranicki

Edinburgh Scotland

Masayuki Yamasaki

Sakado Japan

Decemb er v

Program of the Conference

Surgery and Geometric Top ology

Sakado Saitama

Septemb er Tuesday

Andrew Ranicki

Chain Duality

Erik Pedersen

Simplicial and Continuous Control

Toshiyuki Akita

Cohomology and Euler Characteristics of Coxeter Groups

Septemb er Wednesday

Frank Quinn

Problems with Surgery and Handleb o dies in Low Dimensions

John Ro e

Surgery and Op erator Algebras

Tsuyoshi Kato

The Asymptotic Metho d in the Novikov Conjecture

Eiji Ogasa

On the Intersection of Spheres in a Sphere

Septemb er Thursday

Bruce Hughes

Stratied Spaces and Approximate Fibrations

Francis Connolly

An End Theorem for Manifold Stratied Spaces

TatsuhikoYagasaki

Innitedimensional Manifold Triples of Homeomorphism Groups vii

viii PROGRAM OF THE CONFERENCE

Septemb er Friday

Yong jin Song

The Braid Structure of Mapping Class Groups

Masaharu Morimoto

On Fixed Point Data of Smo oth Actions on Spheres

Masayuki Yamasaki

Epsilon Control and Perl

List of Participants

The numb ers corresp ond to the keys to the photograph on page xi

Toshiyuki Akita Fukuoka Univ akitassatfukuokauacjp

Yuko Asada Nagoya Univ

Naoaki Chinen Univ of Tsukuba

Frank Connolly Univ of Notre Dame USA connollyndedu

Ko ji Fujiwara Keio Univ fujiwaramathkeioacjp

Hiro o Fukaishi Kagawa Univ fukaishiedkagawauacjp

Hiroaki Hamanaka Kyoto Univ

Tetsuya Hosaka Univ of Tsukuba

Bruce Hughes Vanderbilt Univ USA hughesmathvanderbiltedu

Kazumasa Ikeda TokyoUniv nikedahongoeccutokyoacjp

Jun Imai NTT Communication Sci Lab imaicslabkeclnttjp

p Hiroyuki Ishibashi Josai Univ hishimathjosaiacj

Hiroyasu Ishimoto Kanazawa Univ ishimotokappaskanazawauacjp

Osamu Kakimizu Niigata Univ kakimizuedniigatauacjp

Tsuyoshi Kato Kyoto Univ tkatokusmkyotouacjp

Tomohiro Kawakami Osaka Pref College lsipcosakapctacjp

Kazuhiro Kawamura Univ of Tsukuba kawamuramathtsukubaacjp

Takahiro Kayashima Kyushu Univ kayapmathkyushuuacjp

Yukio Kuribayashi Tottori Univ

Yasuhiko Kitada Yokohama National Univ kitadamathlabsciynuacjp

Minoru Kobayashi Josai Univ mkobamathjosaiacjp

Yukihiro Ko dama Kanagawa Univ

Akira Koyama OsakaKyoiku Univ koyamaccosakakyoikuacjp

Kenichi Maruyama Chiba University maruyamacueechibauacjp

Mikiya Masuda OsakaCityUniv masudascihimejitechacjp

Mamoru Mimura Okayama Univ mimuramathokayamauacjp

Kiyoshi Mizohata Josai University mizohatamathjosaiacjp

Masaharu Morimoto Okayama Univ morimotomathemsokayamauacjp

Yutaka Morita Josai Univ ymorieuclidesjosaiacjp ix

x LIST OF PARTICIPANTS

Hitoshi Moriyoshi Hokkaido Univ moriyosimathhokudaiacjp

Mitsutaka Murayama TokyoInstTech murayamamathtitechacjp

Ikumitsu Nagasaki Osaka Univ nagasakimathsciosakauacjp

Masatsugu Nagata Kyoto Univ RIMS nagatakurimskyotouacjp

Kiyoko Nishizawa Josai Univ kiyokomathjosaiacjp

Eiji Ogasa Tokyo Univ imunixccutokyoacjp

Jun OHara Tokyo Metrop olitan Univ oharamathmetrouacjp

Hiroshi Ohta Nagoya Univ ohtamathnagoyauacjp

Nobuatsu Oka Aichi Kyoiku Univ

Kaoru Ono Oc hanomizu Univ onomathochaacjp

Erik Pedersen SUNY at Binghamton USA erikmathbinghamtonedu

Frank Quinn Virginia Tech USA quinnmathvtedu

Andrew Ranicki Edinburgh Univ UK aarmathsedacuk

John Ro e Oxford Univ UK jroejesusoxacuk

Katsusuke Sekiguchi Kokushikan Univ

Kazuhisa Shimakawa Okayama Univ kazumathokayamauacjp

Yong jin Song Inha Univ Korea yjsongmathinhaackr

Toshio Sumi Kyushu Institute of Design sumikyushuidacjp

Toshiki Suzuki Josai Univ

Hideo Takahashi NagaokaInstTech

Junya Takahashi Tokyo Univ

Tatsuru Takakura Fukuoka Univ takakurassatfukuok auacjp

Dai Tamaki Shinshu Univ rivulusgipacshinshuuacjp

Fumihiro Ushitaki Kyoto Sangyo Univ ushitakiksuvxkyotosuacjp

TatsuhikoYagasaki Kyoto Inst Tech yagasakiipckitacjp

Yuichi Yamada Tokyo Univ yyyamadamsutokyoacjp

Hiroshi Yamaguchi Josai Univ hyamamathjosaiacjp

TsuneyoYamanoshita Musashi Inst Tech

Masayuki Yamasaki Josai Univ yamasakimathjosaiacjp

Katsuya Yokoi Univ of Tsukuba yokoimathtsukubaacjp

Zenichi Yoshimura NagoyaInstTech yosimuramathkyy nitechacjp

Surgery and Geometric Top ology

Contents

Foreword v

Program of the conference vii

List of participants ix

Toshiyuki Akita Cohomology and Euler characteristics of Coxeter groups

Frank Connolly and Bogdan Vajiac Completions of StratiedEnds

Zbigniew Fiedorowicz and Yong jin Song The braid structure of mapping class groups

Bruce Hughes Control led topological equivalence of maps in the theory of stratied

spaces and approximate brations

Tsuyoshi Kato The asymptotic method in the Novikov conjecture

Tomohiro Kawakami Anoteonexponential ly Nash G manifolds and vector bund les

Masaharu Morimoto On xedpoint data of smooth actions on spheres

Eiji Ogasa Some results on knots and links in al l dimensions

Erik K Pedersen Control led algebra and topology

Frank Quinn Problems in lowdimensional topology

Andrew Ranicki slides on chain duality

Andrew Ranicki and Masayuki Yamasaki Control led Ltheory preliminary announce

ment

John Ro e Notes on surgery and C algebras

Tatsuhiko Yagasaki Characterizations of innitedimensional manifold tuples and

their applications to homeomorphism groups

COHOMOLOGY AND EULER CHARACTERISTICS

OF COXETER GROUPS

TOSHIYUKI AKITA

Introduction

Coxeter groups are familiar ob jects in many branches of mathematics The

connections with semisimple Lie theory have b een a ma jor motivation for the study

of Coxeter groups Crystallographic Coxeter groups are involved in KacMo o dy

Lie algebras which generalize the entire theory of semisimple Lie algebras Coxeter

groups of nite order are known to be nite reection groups which app ear in

invariant theory Coxeter groups also arise as the transformation groups generated

by reections on manifolds in a suitable sense FinallyCoxeter groups are classical

ob jects in combinatorial group theory

In this pap er we discuss the cohomology and the Euler characteristics of nitely

generated Coxeter groups Our emphasis is on the role of the parab olic subgroups

of nite order in b oth the Euler characteristics and the cohomology of Coxeter

groups

The Euler characteristic is dened for groups satisfying a suitable cohomological

niteness condition The denition is motivated by top ology but it has applications

to group theory as well The study of Euler characteristics of Coxeter groups was

initiated by JP Serre who obtained the formulae for the Euler characteristics

of Coxeter groups as well as the relation b etween the Euler characteristics and the

Poincare series of Coxeter groups The formulae for the Euler characteristics of

Coxeter groups were simplied byIM Chiswell From his result one knows

that the Euler characteristics of Coxeter groups can b e computed in terms of the

orders of parab olic subgroups of nite order

On the other hand for a Coxeter group W the family of parab olic subgroups of

nite order forms a nite simplicial complex F W In general given a simplicial

complex K the Euler characteristics of Coxeter groups W with F W K are

b ounded but are not unique However it follows from the result of M W Davis

nsphere Theorem Inspired that eW if F W is a generalized homology

by this result the author investigated the relation b etween the Euler characteris

tics of Coxeter groups W and the simplicial complexes F W and obtained the

following results

If F W is a PLtriangulation of some closed nmanifold M then

If F W is a connected graph then eW F W where denotes

the genus of the graph

TOSHIYUKI AKITA

See Theorem and Conversely given a PLtriangulation K of a closed n

manifold M we obtain an equation for the number of isimplices of K i n

by considering a Coxeter group W with K F W Theorem and its corollary

The family of parab olic subgroups of nite order is also imp ortant in understand

ing the cohomology of a Coxeter group W For instance let k b e a commutative

ring with unity a ring homomorphism

H Wk H W k

induced by restriction maps where W ranges all the parab olic subgroups of nite

order Then u ker is nilp otent and cannot b e detected b yany nite subgroup

of W And wecansay more ab out the homomorphism

We remark that according to the results of D Quillen and K S Brown

the family of elementary ab elian psubgroups also plays an imp ortant role

However it is plo cal The role of the parab olic subgroups of nite order is not

plo cal a phenomenon in which I am very interested

Notation For a nite set X the cardinalityof X is denoted by jX j In partic

ular for a nite group G the order of G is denoted by jGj

Definitions and Examples

In this section wegive the denition and elementary examples of Coxeter groups

Denition Let S b e a nite set Let m S S N fg b e a map satisfying

the following three conditions

ms tmt s for all s t S

ms s for all s S

ms t for all distinct s t S

The group W dened by the set of generators S and the fundamental relation

mst

s t ms t is called a Coxeter group Some authors p ermit S to

b e an innite set

Sm instead of W to emphasize S Remark We frequently write WSorW

and m The pair WS is sometimes called a Coxeter system in the literature

Remark Each generator s S is an element of order in W Hence W is

generated byinvolutions

Example Let WSbeaCoxeter group with S fs tg If ms t then

W is isomorphic to D the dihedral group of order ms t If ms t

mst

then W is isomorphic to ZZ ZZ the free pro duct of two copies of the cyclic

group of order

Example A nite reection group is a nite subgroup of the orthogonal group

O n for some n generated by orthogonal reections in the Euclidean space A

nite reection group is known to b e a Coxeter group ie it admits a presentation

of Coxeter groups ConverselyanyCoxeter group of nite order can b e realized as

a nite reection group Hence one can identify Coxeter groups of nite order with

nite reection groups in this way

For example an elementary ab elian group ZZ and a symmetric group

can be regarded as Coxeter groups Finite reection groups are completely n

COXETER GROUPS

classied and their list is short By using the list it is easy to determine if a given

Coxeter group is of nite order See for details

Example Coxeter groups are closed under free pro ducts and direct pro ducts

Example Full triangular group Let p q r be integers greater than The

group T p q r dened by the presentation

p q r

T p q r s s s js s s s s s s

is called the ful l triangular group It is obvious from the presentation that T p q r

is a Coxeter group The group T p q r isknown to b e of nite order if and only

p q r

The triangular group

p q r

T p q r uvju v uv

is a subgroup of T p q r of index via u s s and v s s

The full triangular group T p q r can be realized as a planar discontinuous

group acting on a sphere S if p q r on the Euclidean plane E

if p q r or on the hyp erb olic plane H if p q r

The orbit space of the action of T p q r onS E orH is homeomorphic to a

disk D

Example Given integers p q r greater than let O p q r be the orbifold

dened as follows See for the notion of orbifolds The underlying space of

O is a standard simplex Vertices v v and v of are corner reection

points of order p q and r The p oints in the interior of edges are reection

points while the p oints in the interior of the whole are manifold p oints

The orbifold O p q r is uniformable ie it has a manifold cover Indeed the

action of the full triangular orbifold O p q r comes from the orbit space of the

group T p q r ononeofS E orH mentioned in the b ottom of Example

or b

The orbifold fundamental group O p q r is isomorphic to T p q r

Let O p q r be the orbifold whose underlying space is a sphere S with

three cone p oints of order p q and r Then there is a double orbifold covering

O p q r O p q r

or b

The orbifold fundamental group O p q r is isomorphic to the triangular

group T p q r See and for the details

Example Example and are sp ecial cases of reection orbifolds and

groups generated by reections on a manifold b oth of which are closely related to

Coxeter groups See and for the general theory

Parabolic Subgroups

Let WSm be a Coxeter group For a subset T S dene W to be the

subgroup of W generated by the elements of T ie W T W In

particular W fg and W W W is called a parabolic subgroup or sp ecial

S T

subgroup of W The subgroup W is known to be a Coxeter group Indeed

W TmjT T is a Coxeter group It is obvious from the denition that the

numb er of parab olic subgroups of a Coxeter group is nite

TOSHIYUKI AKITA

Example Parab olic subgroups of the full triangular group T p q r consist

of subgroups Namely

The trivial subgroup fg

Three copies of a cyclic group of order generated by single element

Dihedral groups of order pq andr generated bytwo distinct elements

T p q r itself

The following observation asserts that the parab olic subgroups of nite order are

maximal among the subgroups of nite order in a Coxeter group

Prop osition Lemma Let W be a Coxeter group and H its nite sub

group Then thereisaparabolic subgroup W of nite order and an element w W

such that H wW w

Euler characteristics

In this section weintro duce the Euler characteristics of groups First weintro

duce the class of groups for which the Euler characteristic is dened

Notation Let b e a group Then Zistheintegral group ring of W e regard

Z as a Zmo dule with trivial action

Denition A group is said to b e of type FL if Z admits a free resolution

over Z of nite typ e In other words there is an exact sequence

F F F F Z

n n

of nite length such that each F is a nitely generated free Zmo dule

Remark If is a group of typ e FL then cd and hence is torsionfree

Denition A group is said to be of type VFL if some subgroup of nite

index is of typ e FL

Nowwe dene the Euler characteristic of a group Let b e a group of typ e FL

and let

F F F F Z

n n

b a free resolution of nite length The Euler characteristic e of is dened by

rank F e

Z i

Let b e a group of typ e VFL Then its Euler characteristic e is dened by

e Q

where is a subgroup of nite index which is of typ e FL The rational number

e is indep endent of the choice of a subgroup andwehave

Prop osition Let beagroup and a subgroup of nite index Then is of

type VFL if and only if is of type VFL If is of typeVFLthen

e e

Wegive some examples of groups of typ e VFL and their Euler characteristics

COXETER GROUPS

Example Any nite group is of typ e VFL Its Euler characteristic is given

Take to b e a trivial group fg

Example Let K be a nite aspherical p olyhedron Then its fundamental

group K isoftyp e FL and

e K

where K istheEulercharacteristic of K The fact that the Euler characteristic

of a nite aspherical p olyhedron dep ends only on its fundamental group is the

motivation of the denition of Euler characteristics of groups

For instance the circle S is aspherical and S Z hence

eZ S

Let b e a closed orientable surface of genus g Then is aspherical proving

g g

e g

g g

Example If are groups of typ e VFL then their free pro duct

and their direct pro duct are of typ e VFL and

e e e

As a consequence a free group F and a free ab elian group Z are of typ e VFL in

fact typ e FL and wehave

eF n

where F is the free group of rank n

Example The group SL Z has a subgroup of index which is isomorphic

to the free group of rank Hence SL Zisoftyp e VFL Using Example and

one can compute the Euler characteristic of SL Zas

eSL Z

Example The Euler characteristics of groups are closely related to the Euler

characteristics of orbifolds See or for the denition of the orbifold Euler

characteristics Namely let O b e an orbifold such that

O has a nite manifold covering M O for which M has the homotopytyp e

of a nite complex

The universal cover of O is contractible

or b

Then the orbifold fundamental group O ofO is of typ e VFL and one has

or b

e O

or b

where O is the orbifold Euler characteristic of O

TOSHIYUKI AKITA

Example Let b e a full triangular group T p q r of innite order Then

as in Example is isomorphic to the orbifold fundamental group of the orbifold

O p q r The orbifold O p q r satises the conditions and in Example

Hence the Euler characteristic of is identied with the orbifold Euler characteristic

of O p q r Using this one has

p q r

Finallywemention two prop erties of Euler characteristics of groups Let G be

a group of typ e VFL

Theorem GottliebStallings If eG then the center of G is a

nite subgroup

Theorem Brown Let p be a prime If p divides the denominator of eG

then G has a subgroup of order p

In view of Example Theorem is a generalization of a part of Sylows theorem

Euler characteristics of Coxeter groups I

JP Serre proved that Coxeter groups are of nite homological typ e In

xeter groups satisfy a much stronger niteness condition fact he proved that Co

than nite homological typ e called typ e WFL He also provided the formulae for

the Euler characteristics of Coxeter groups

The formulae of Euler characteristics of Coxeter groups were simplied byIM

Chiswell whichwenow quote Before doing this we remark that if a Coxeter

group W is of nite order then its Euler characteristic is given by eW jW j

Example Hence wemayassumea Coxeter group W to b e of innite order

Theorem Chiswell The Euler characteristic eW of a Coxeter group W

of innite order is given by

X X

jT j jT j

eW eW

jW j

T S T S

jW j jW j

T T

Thus the Euler characteristics of Coxeter groups are completely determined their

parab olic subgroups of nite order Since the order of a nite reection group is

easy to compute so is the Euler characteristic of a Coxeter group

Serre also obtained in the relation between the Euler characteristic of a

Coxeter group and the Poincare series Namely for a Coxeter group WS dene

t g t

w W

where l w is the minimum of the length of reduced words in S representing w

The function g t is known to b e a rational function and is called Poincare series

Serre proved of WS

In general Poincare series of arbitrary nitely presented groups may not satisfy

this prop erty See

COXETER GROUPS

Poset of Parabolic Subgroups of Finite Order

Before continuing the discussion of the Euler characteristics of Coxeter groups

we intro duce the simplicial complexes asso ciated with Coxeter groups Given a

Coxeter group WS dene F W to be the poset of nontrivial subsets F S

such that the order of the corresp onding parab olic subgroup W is of nite order

If there is no ambiguity we write F instead of F W The poset F W can be

regarded as an abstract simplicial complex with the set of vertices S

Example If WS is a nite reection group with jS j nthenany nontrivial

subset F S b elongs to F W and hence

the standard n simplex

Example If WS is a full triangular group of innite order then

the b oundary of the standard simplex ie a triangle

Example The list of Coxeter groups with F W can b e found in

Example Let K be a nite simplicial complex A nite simplicial complex

K is called a ag complex if K satises the following condition For any subset

V fv v g of vertices of K ifanytwo element subset fv v g of V form an

n i j

edge of K then V fv v g spans an nsimplex A barycentric sub division

Sd K of a nite simplicial complex K is an example of a ag complex

If K is a ag complex then there is a Coxeter group W for which F W K

NamelyletS b e the set of vertices of K Dene m S S N fg by

s s

ms s

fs s g forms a simplex

otherwise

The resulting Coxeter group WS satises F W K In particular given a

nite simplicial complex K thereisaCoxeter group W with F W SdK

Denition A Coxeter group WS with all ms t or for distinct

s t S is called rightangled Coxeter group

Coxeter groups constructed in Example are examples of rightangled Coxeter

groups Conversely if W is a rightangled Coxeter group then F W is a ag

complex

Remark It is not known if there is a Coxeter group W for which F W K for

agiven nite simplicial complex K

Euler characteristics of Coxeter groups I I

Nowlet us consider the Euler characteristic of W in terms of the structure of

F W Pro ofs of statements of the following three sections will app ear in If

WS is a nite reection group then

jS j

jW j

TOSHIYUKI AKITA

The equality holds if and only if W is isomorphic to the elementary ab elian group

jS j

ZZ of rank jS j Nowlet K b e a nite simplicial complex Then the Euler

characteristic of anyCoxeter group W with K F W must satisfy

X X

f K f K

i i

iev en

iodd

where f K isthenumber of isimplices of K This follows from Theorem and the

equation As in the following example the inequality is not b est p ossible

Example Supp ose K Then Coxeter groups W with F W K are

precisely full triangular groups of innite order The inequality implies

On the other hand from the formula in Example one has

which is b est p ossible

Example shows that for a xed nite simplicial complex K Euler characteristics

of Coxeter groups with F W K can vary However from the result of M W

Davis one has

Theorem Let W be a Coxeter group such that F W is a generalized homology

nsphere then

Here a generalized homology nsphere is a simplicial complex K satisfying

K has the homology of a nsphere

The link of an isimplex of K has the homology of a n i sphere

A simplicial complex satisfying the condition and is also called a Cohen

Macaulay complex A triangulation of a homology sphere is an example of a gener

alized homology sphere

Note that Davis actually proved that if W is a Coxeter group suchthat F W

is a generalized homology nsphere then for each torsion free subgroup of nite

index in W there is a closed aspherical n manifold M with M

Theorem It follows that

e M

W W

since M is o dd dimensional and has homotopytyp e of a nite simplicial complex

We partially generalize Theorem A nite simplicial complex K is a PL

triangulation of a closed M if for each simplex T of K the link of T in K is

a triangulation of dim M dim T sphere If K is a PLtriangulation of a

homology sphere then K is a generalized homology sphere

Theorem T Akita Let W be a Coxeter group such that F W is a PL

triangulation of a closed nmanifold then

F W

where F W is the Euler characteristic of the simplicial complex F W

COXETER GROUPS

Remark Given a simplicial complex K there is a Coxeter group W such that

F W agrees with the barycentric sub division of K Example Hence there

are Coxeter groups for which Theorem and Theorem can b e applied

Remark We should p oint out that the assumptions of Theorem and p ermit

for instance K to be an arbitrary triangulation of nsphere The signicance

b ecomes clear if we compare with the case that K is a triangulation of a circle S

Indeed the Euler characteristics of Coxeter groups with F W a triangulation of a

circle S can b e arbitrary small

Remark Under the assumption of Theorem eW is an integer On the

other hand given a rational num ber q there is a Coxeter group W with eW q

Application of Theorem

Let K be a ag complex Let WS be a Coxeter group with F W K as

in Example Any parab olic subgroup W of nite order is isomorphic to the

jF j

elementary ab elian group ZZ of rank jF j Hence the Euler characteristic

of W is determined bythenumb er of simplices of K Explicitlyletf K bethe

number of isimplices of K Then

f K eW

Using this together with Theorem one obtains

Theorem T Akita Let K be a PLtriangulation of a closed nmanifold As

sume K is a ag complex Then

K f K

In particular the barycentric sub division of any nite simplicial complex is a ag

complex Thus

Corollary Let K be a PLtriangulation of a closed nmanifold Let f Sd K is

the number of isimplices of the barycentric subdivision Sd K of K Then

Sd K f Sd K

In general if K is a triangulation of a closed nmanifold then

K f K

f K f K

n n

hold The equality in Theorem is not the consequence of the equalities

For a triangulation K of a sphere S for arbitrary n the DehnSommerville

equations give a set of equations for the f K s It would be interesting to in

vestigate the relation between the equation in Theorem and DehnSomerville equations

TOSHIYUKI AKITA

Euler Characteristics of Aspherical Coxeter Groups and the

Genus of a Graph

In this section we consider the Euler characteristics of Coxeter groups W such

that F W is a graph dimensional simplicial complex

Denition ACoxeter group WSiscalledaspherical in if every three

distinct elements of S generate a parab olic subgroup of innite order

In view of Example a Coxeter group WS is aspherical if and only if for every

three distinct elements s t u S

m m m

st tu us

holds where by the convention It is easy to see that a Coxeter group W

is aspherical if and only if F W is a graph

For a graph let E b e the set of edges of When WS is an aspherical

Coxeter group with F W it follows from Chiswells formula that

jS j jS j jE j

One has another inequalityforeW using the genus of a graph The genus of a

graph denoted by is the smallest number g such that the graph imbeds in

the closed orientable surface of genus g For instance a graph is a planar graph

if and only if

Theorem T Akita Let WS be a Coxeter group for which F W is a con

nected nite graph Then

eW F

Example For any nonnegativeinteger n there is a Coxeter group W satis

fying

F W is a graph of genus n

eW n

The construction uses the complete bipartite graphs K

Recall that a graph is a bipartite graph if its vertex set can b e partitioned into

twosubsets U and V such that the vertices in U are mutually nonadjacent and the

vertices in V are mutually nonadjacent If every vertex of U is adjacenttoevery

vertex of V then the graph is called completely bipartite on the sets U and V A

complete bipartite graph on sets of m vertices and n vertices is denoted by K

Now the genus of the completely bipartite graph K is given by

m n

See Theorem Now let S S t S with jS j m jS j n Dene

m S S N fg by

s t

ms t

s S t S i j

i j

otherwise

COXETER GROUPS

Then the resulting Coxeter group WS is rightangled and satises F W K

Its Euler characteristic is given by

m n

Alternatively one can construct similar examples by using complete graphs

Cohomology of Coxeter Groups

In this section we are concerned with the cohomology of Coxeter groups The

content of this section extends the earlier pap ers and We restrict our atten

tion to the relation b etween the cohomology of Coxeter groups and the cohomology

of parab olic subgroups of nite order Let WS be a Coxeter group Let k be

acommutative ring with identity regarded as a W mo dule with trivial W action

Set

H Wklim H W k

W F

where W runs all p ossibly trivial parab olic subgroups of nite order The inverse

limit is taken with resp ect to restriction maps H W H W for F F

Let

H Wk H Wk

b e the ring homomorphism induced by the restriction maps H Wk H W k

The prop erties of are the main topic of this section

D J Rusin M W Davis and T Januszkiewicz computed the mo d

cohomology ring of certain Coxeter groups

Theorem Corollary Let W be a Coxeter group with hyperbolic signa

ture with al l rank parabolic subgroups hyperbolic and with even exponents ms t

Then

H W F F u w rst S

r st

with relations u w if r s and r t w w unless frsg ft ug

r st rs tu

and u u if divides ms t but u u w otherwise

s t s t st

Here we shall not explain the assumptions in Theorem Instead we p oint out that

if all ms t with s t distinct are large enough compared with the cardinality S

then the resulting Coxeter group has hyp erb olic signature and its rank parab olic

subgroups are hyp erb olic SuchaCoxeter group must b e aspherical

Theorem Theorem Let W be a rightangled Coxeter group Then

v v I H W F F

where I is the ideal generated by al l square free monomials of the form v v

i i

whereatleast two of the v do not commute when regarded as elements of W

See Denition for the denition of rightangled Coxeter group From their

results one can show that induces an isomorphism

H W F H W F

for a Coxeter group W which satises the assumptions in Theorem or Inspired

by this observation weprov ed

TOSHIYUKI AKITA

Theorem Let W be a Coxeter group and k acommutative ring with identity

Let H Wk H Wk be as above Then the kernel and the cokernel of

consist of nilpotent elements

A homomorphism satisfying these prop erties is called an Fisomorphism in

Notice that unlike the famous result of Quillen concerning of the mo d p co

homology of groups of nite virtual cohomological dimension the co ecient ring k

can b e the ring Z of rational integers

Example Let W be the full triangular group T Its mo d coho

mology ring is given by H W F F u v u where deg u and deg v

p while H W F is isomorphic to F w withdeg w Then u

and hence has nontrivial kernel for k F This shows the homomorphism may

not b e an isomorphism in general

Unfortunatelywedonotknow whether mayhave a nontrivial cokernel We

give a sucient condition for to b e surjective

Theorem Suppose that W is an aspherical Coxeter group see Denition

Then is surjective for any abelian group A of coecients with trivial W action

groups satisfying the assumptions in Theorem must be For example Coxeter

aspherical

In the case k F there is more to say By Theorem the homomorphism

p p p

induces the homomorphism H Wk H Wk where denotes

nilradical Rusin proved that the mo d cohomology ring of any nite Coxeter

group nite reection group has no nilp otent elements Theorem Hence

the nilradical of H W F is trivial From this together with Theorem and

we obtain

Corollary For any Coxeter group W induces a monomorphism

H W F H W F

which is an isomorphism if W is aspherical

Remark Another study of the relation between the cohomology of aspherical

Coxeter groups and their parab olic subgroups of nite order can b e found in

Now we turn to our attention to detection by nite subgroups An element

u H Wkissaidtobedetected by nite subgroups if the image of u by the map

Y Y

res H Wk H H k

H H

is nontrivial where H runs all the nite subgroups of W It would b e of interest to

knowwhich elements of H Wk H Wk are detected by nite subgroups One

can reduce this question to the following prop osition which follows from Theorem

and Prop osition

Prop osition An element u H Wk is detected by nite subgroups if and

only if u ker

Finallywegive a example of elements of H Wkwhich cannot b e detected by nite subgroups

COXETER GROUPS

Example Let W be the full triangular group T Its mo d coho

mology ring is given by in Example One can check easily that uv n

is contained in ker Thus uv n cannot b e detected by nite subgroups as

elements of H W F

Remark The virtual cohomological dimension of anyCoxeter group W is known

to be nite p and its FarrellTate cohomology written H Wk is

dened For the FarrellTate cohomology the analogues of Theorem and and

Prop osition are valid See and for detail

Outline of Pr oof

Actions of Coxeter groups A suitable complex on whichaCoxeter group

acts is used in the pro of of Theorem We recall how this go es Let WSbea

Coxeter group Let X b e a top ological space X b e a family of closed subsets

s sS

of X indexed by S From these data one can construct a space on which W acts

as follows Set

S xfs S x X g

and let U U X W X W b eing discrete where the equivalence relation

is dened by

w x w x x x w w W

S x

Then W acts on U X by w w xw w x The isotropy subgroup of w xis

wW w

S x

Pro of of Theorem Outline Given a Coxeter group WS let X

b e the barycentric sub division of c FW the cone of F W with the cone p oint

c Dene X to b e the closed star of s S here s S is regarded as a vertex of

F W and hence a vertex of X Then one of the main results of M W Davis

x asserts that U X is contractible

Consider the sp ectral sequence of the form

q pq

H W k H Wk E

is isomorphic to H Wk and In the sp ectral sequence one can provethat E

the homomorphism is identied with the edge homomorphism H Wk E

Observe that

E if p

There is a natural number nsuchthatn E for all p and q

Together with these observations Theorem follows from the formal prop erties

of the dierentials of the sp ectral sequence

References

T Akita On the cohomology of Coxeter groups and their nite parab olic subgroups Tokyo

J Math

On the cohomology of Coxeter groups and their nite parab olic subgroups II

preprint

Euler characteristics of Coxeter groups PLtriangulations of closed manifolds and

the genus of the graph in preparation

N Bourbaki Groupes et algebres de Lie Chapitres et Hermann Paris

TOSHIYUKI AKITA

K S Brown Euler characteristics of groups The pfractional part Invent Math

Cohomology of Groups Graduate texts in mathematics vol SpringerVerlag

New YorkHeidelb ergBerlin

I M Chiswell The Euler characteristic of graph pro ducts and of Coxeter groups Discrete

groups and geometry Birmingham London Math Soc Lecture Notes Cambridge

University Press Cambridge

M W Davis Groups generated by reections and aspherical manifolds not covered by Eu

clidean space Ann of Math

Some aspherical manifolds Duke Math J

TJanuszkiewicz Convex p olytop es Coxeter orbifolds and torus actions Duke Math

W J Floyd S P Plotnick Growth functions on Fuchsian groups and the Euler character

istics Invent Math

D H Gottlieb A certain subgroup of the fundamental group Amer J Math

J L Gross T W Tucker Topological Graph Theory A WileyInterscience Publication John

Wiley Sons New York

R B Howlett On the Schur multipliers of Coxeter groups J London Math Soc

J E Humphreys Reection groups and Coxeter groups Cambridge studies in advanced

mathematics vol Cambridge University Press Cambridge

M Kato On uniformization of orbifolds Homotopy Theory and Related Topics Advanced

Studies in Pure Math

J Milnor On the dimensional Briesk orn manifolds M p q r in Knots groups and

manifolds Annals of Math Studies

S J Pride R Stohr The cohomology of aspherical Coxeter groups J London Math Soc

D Quillen The sp ectrum of an equivariant cohomology ring I I I Ann Math

and

D J Rusin The cohomology of groups generated by involutions Ph D thesis Chicago

University

P Scott The geometry of manifolds Bul l London Math Soc

JP Serre Cohomologie de group es discrets Ann of Math Studies vol Princeton

University Press Princeton

J Stallings Centerless groupsan algebraic formulation of Gottliebs theorem Topology

W Thurston The geometry and top ology of manifolds Princeton University preprint

Department of Applied Mathematics Fukuoka University Fukuoka Japan

Email address akitassatfukuokauac jp

COMPLETIONS OF STRATIFIED ENDS

FRANK CONNOLLY AND BOGDAN VAJIAC

Introduction

A famous result of L Sieb enmann characterizes those top ological manifolds

which are the interiors of compact manifolds with b oundary Elsewhere we have

recently shown that his theorem generalizes to the context of stratiedspaces Our

purp ose here is to explain the main results of our work briey See for the full

account

Definitions

Let X n b e a tame ended top ological nmanifold Sieb enmann proves that

there is a single obstruction X in K Z with the prop erty that X if

and only if X is the interior of a compact manifold with b oundary By the work of

Freedman and Quinn one can also allow n if is not to o complicated The

group denotes the fundamental group of the end of X which can b e describ ed

b b

as the HolinkX here X denotes the onep oint compactication of X The

space HolinkX A the homotopy link of A in X is dened for any subspace

A of a top ological space X as

H ol ink X Af Map X j Ag

It is given the compactop en top ology It comes with twomaps

j p

X X

X A j H ol ink X A A p

X X

It is used by Quinn as a homotopical analogue for the normal sphere bundle

of A in X

F Quinn generalizes Sieb enmanns result greatly For any lo cally compact pair

X A where A is closed and tame in X X A is an nmanifold n and A

A p which is an ANR Quinn denes an obstruction q X A K

vanishes if and only if A has a mapping cylinder neighb orho o d in X Here the map

p HolinkX A A is the pro jection This concept of tameness is discussed by

many others at this conference The foundational concepts surrounding controlled

K theory have recently b een greatly claried bythe eminently readable pap er of

Ranicki and Yamasaki

X A can b e lo calized in the following way let A be Quinns obstruction q

a closed and tame subset of X and X an op en subset of X Then A X A

lf lf

is tame in X and i q X Aq X A wherei K A p K A p is

X X

the restriction map Using these maps one can dene for every subset B A

lf lf

K A p lim K A p jA

X X

First author partially supp orted by NSF Grant DMS

FRANK CONNOLLY AND BOGDAN VAJIAC

the direct limit is over the X neighborhoods X of B Then the image of

any one of the obstructions q X A in K A p is indep endent of the

X neighborhood X chosen We will write this image q X A

In geometric top ology the stratied analogue of a top ological manifold is a

stratiedspace This concept was intro duced by Quinn under the name manifold

b erger homotopy stratied set our terminology is due to Hughes and Wein

A stratied space is a lo cally compact space nitely ltered by closed subsets

n n i i

X X X X Each stratum X X X must be a

manifold and the boundary of X dened by the rule X X must be

i i

closed in X It is customary to arrange the indexing so that dimX i It is

also required that X must be tame in X X for eac h j i The pro jection

i i j

HolinkX X X X must b e a bration and the inclusion HolinkX

i j i i i

X X HolinkX X X j must b e a b er homotopy equivalence over

j i i j i X

Main results

Let X b e a stratied space with empty b oundary We seek a completion of X

ie a compact stratied space X such that X X X and X has a collar

neighborhood in X It is easy to see that a necessary condition for X having a

completion is to b e tame ended This means that the one p oint compactication

of X X X is again a stratied space The stratication of the onep oint

compactication is the following

b b

X X X X j

j j

An equivalentformulation is that fg is tame in X fg for each j Notice that

byreverse tameness each X can have only nitely manyends

A completion may not always exist a weaker requirementwould b e an exhaustion

of X This is dened to b e an increasing sequence of compact stratied subspaces

whose union is X An exhaustion is also of X with bicollared b oundaries in X

obstructed in the category of stratied spaces

Our main results and say that a completion or an exhaustion

exists if a single obstruction vanishes

The End Obstruction Let X be a tame ended stratied space For each

integer mwe dene as in ab ove

m m m

b b b

X p X q X X K

m m m

b b b

As b efore the map p H ol ink X X X denotes the Holink pro

jection Set also

X X K X p

m m

Theorem Suppose X is a stratied space with empty boundary which ad

mits a completion Then X

Theorem Let X be a tame endedstratiedspace with empty boundary Let

ontaining X suchthatA admits a completion A be any closedpure subset of X c

AA A Suppose X Then X admits a completion X such that Cl

COMPLETIONS OF STRATIFIED ENDS

A pure subset is one which is the union of comp onents of strata

Note This result reduces to Siebenmanns theorem when X has only one

stratum

Note Following Weinb erger wesay that a nite group action on a manifold

M Gisastratied G manifold if the xed set of each subgroup M is a manifold

H K

and M is lo cally at in M for each K H By and of this

is equivalenttosaying that X M G is a stratied space when it is stratied by

its orbit typ e comp onents A corollary of our main theorem is an endcompletion

result for Gmanifolds

Corollary Let M G bea stratied Gmanifold with M Then M G

is the interior of a compact stratied Gmanifold with col lared boundary iff X

M G is tame ended X andX has a completion

The obstruction to nding an exhaustion for the stratied space X turns out

to have the form X where is a map we will not dene here in com

plete generality Instead we will give the denition of X in the sp ecial case

n n

when X admits a completion In this case an neighb orho o d in X has

the form B for some stratied space B Then the open cone of B

OB which can be thought of as B B is a neighborhood of in

X moreover has a conal sequence of suchneighborhoods B k B

k The restriction maps connecting the K theory of these are

that the obstruction X reduces to X and isomorphisms This implies

lf lf

OB p j where X p can b e identied to K moreover that K

b b

X X

p j

b b

HolinkX X j OB is the pro jection map The inclusion map induces

a restriction map

lf lf

K OB p K B pj

whichamounts then to a map

K X p K B p

n B

where p denotes the restriction of the holink pro jection over B

This is the map we seek It turns out that X K B p is the ob

n n B

struction to nding an exhaustion of X

Theorem Exhaustibility Theorem Let X be a tame ended ndimensional

stratiedspace with empty boundary for which X admits a completion Assume

that X Then X admits an exhaustion

Conversely if X admits an exhaustion and al l the fundamental groups of the

n n

bers of the map H ol ink X X X are good then X

Wesay a group G is good if K ZG for i

No example of a group which is not go o d is known Moreover a recent theorem of

Farrell and Jones shows that any subgroup of a uniform discrete subgroup of a

virtually connected Lie group must b e go o d

There are stratied Gmanifolds which are not exhaustable but are tame

Z on M R fgn ended In fact there is a semifree action of G Z

with xed set R fgforwhich M G in K ZG K ZG

K ZG K ZG Furthermore if M S S and M M is the

FRANK CONNOLLY AND BOGDAN VAJIAC

usual covering map then is equivariant with resp ect to a stratied Gaction

on M This Gmanifold M G is hcob ordant stratied and equivariant to

some M G whose innite cyclic cover M G has the form V fgG where

V G is a linear representation of G This example and the more general question

of realizability of the obstruction X are thoroughly analyzed in the PhD

thesis of B Vajiac

References

FX Connolly and B Vajiac An End Theorem for StratiedSpaces preprint pp

Available byanonymous FTP from mathuiucedupubpap ersKtheory

F T Farrell and L E Jones Isomorphism conjectures in algebraic Ktheory J Amer Math

So c

M Freedman and F Quinn Topology of Manifolds Princeton Mathematical Series Princeton

University Press Princeton N J

F Quinn Ends of maps I Ann of Math

Ends of maps IIInvent Math

Homotopical ly stratiedsets J Amer Math So c

A A Ranicki and M Yamasaki Control led K theoryTop ology Appl

Department of Mathematics University of Notre Dame Notre Dame IN USA

Email address FrancisXConnollynd edu BogdanEVajiacnde du

THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

Zbigniew Fiedorowicz and Yongjin Song

Intro duction

It was shown by Stashe and MacLane that monoidal categories give rise to

lo op spaces A recognition principle sp ecies an internal structure such that a space X has

such a structure if and only if X is of the weak homotopy typ e of nfold lo op space It

has b een known for years that there is a relation b etween coherence problems in homotopy

theory and in categories Mays recognition theorem states that for little ncub e op erad

C n every nfold lo op space is a C space and every connected C space has the weak

n n n

homotopytypeofannfold lo op space

E Miller observed that there is an action of the little square op erad on the disjoint

union of BDif f S s extending the F pro duct which is induced by a kind of connected

sum of surfaces We hence have that the group completion of q BDif f S is a double

g g

lo op space up to homotopy Miller applied this result to the calculation of the homology

groups of mapping class groups However his description of the action of the little square

op erad is somewhat obscure On the other hand the rst author proved that the group

completion of the nerve of a braided monoidal category is the homotopytyp e of a double lo op

space This result implies that there exists a strong connection b etween braided monoidal

category and the mapping class groups in view of Millers result

We in this pap er show that the disjoint union of s is a braided monoidal category

with the pro duct induced by the connected sum Hence the group completion of q B

g g

is the homotopy typ e of a double lo op space We explicitly describ e the braid structure

of q regarding as the subgroup of the automorphism group of S that

g g g g

consists of the automorphisms xing the fundamental relator Weprovide the formula for

the braiding Lemma which is useful in dealing with the related problems Using this

braiding formula we can make a correction on Cohens diagram We also showthatthe

double lo op space structure of the disjoint union of classifying spaces of mapping class groups

cannot b e extended to the triple lo op space structure Theorem It seems imp ortant

to note the relation b etween the braid structure and the double lo op space structure in an

explicit way

Turaev and Reshetikhin intro duced an invariant of ribb on graphs which is derived from

the theory of quantum groups and is a generalization of Jones p olynomial This invariantwas

extended to those of manifolds and of mapping class groupscf The denitions

are abstract and a little complicated since they are dened through quantum groups G

The second author was partially supp orted by GARCKOSEF

ZBIGNIEW FIEDOROWICZ AND YONGJIN SONG

Wright computed the ReshetikhinTuraev invariant of mapping class group explicitly

in the case r that is at the sixteenth ro ot of unity For each h we can nd

the corresp onding colored ribb on graph whose ReshetikhinTuraev invariant turns out to

k k k k

g g

be an automorphism of the dimensional summand of V V V V which

we denote by V We get this ribb on graph using the Heegaard splitting and the surgery

theory of manifolds Wright showed as a result of her calculation that the restriction

of this invarian t to the Torelli subgroup of is equal to the sum of the BirmanCraggs

g g

homomorphisms dimV so the ReshetikhinTuraev invariantof h

g g

g g g g

when r isa matrix with entries of complex numbers Wright

proved a very interesting lemma that there is a natural onetoone corresp ondence b etween

the basis vectors of V and the Zquadratic forms of Arf invariant zero It would be

interesting to check if the ReshetikhinTuraev representation preserves the braid structure

Mapping class groups and monoidal structure

Let S b e an orien table surface of genus g obtained from a closed surface by removing

k op en disks The mapping class group is the group of isotopy classes of orientation

preserving selfdieomorphisms of S xing the b oundary of S that consists of k disjoint

gk gk

circles Let Dif f S b e the group of orientation preserving selfdieomorphisms of S

gk gk

We also have the following alternative denition

Dif f S

gk o gk

We will mainly deal with the case k and k and are generated byg

g g

Dehn twistscf There is a surjective map

g g

Figure Dehn twists

Many top ologists are interested in the homology of mapping class groups An interesting

observation is that there is a pro duct on the disjoint union of Dif f S s It is known by

Stashe and MacLane that if a category C has a monoidal structure then its clas

sifying space gives rise to a space which has the homotopytyp e of a lo op space Fiedorowicz

THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

showed that a braid structure gives rise to a double lo op space structure Wenow recall

the denition of strict braided monoidal category

Denition A strict monoidal or tensor category C E is a category C together

with a functor CCC called the product and an ob ject E called the unit object

satisfying

a is strictly asso ciative

b E is a strict sided unit for

Denition A monoidal category C E iscalledastrict braided monoidal category

if there exists a natural commutativity isomorphism C AB B A satisfying

c C C

AE EA A

d The following diagrams commute

ABC

AB C C AB

C C

A BC AC B

AC B

AB C

B C A AB C

C C

AB C B AC

B AC

The rst author recently gave a pro of of the following lemma

Lemma The group completion of the nerve of a braided monoidal category is the

homotopy type of a double loop space The converse is true

Miller claimed in that there is an action of the little square op erad of disjoint squares

in D on the disjoin t union of the B s extending the F pro duct that is induced bythe

connected sum Here the F pro duct is obtained byattaching a pair of

g h g h

pants a surfaces obtained from a sphere byremoving three op en disks to the surfaces S

and S along the xed b oundary circles and extending the identity map on the b oundary

to the whole pants Hence according to Mays recognition theorem on the lo op spaces

the group completion of q B is homotopy equivalent to a double lo op space Millers

g g

prop osition seems correct although the details are not so transparent In view of lemma

the disjoint union of s should b e related to a braided monoidal category Here we

regard q as a category whose ob jects are g g Z and morphisms satisfy

g g

if g h

g h Hom

if g h

ZBIGNIEW FIEDOROWICZ AND YONGJIN SONG

Without sp eaking of the action of little square op erad we are going to show that the

group completion of q B is homotopy equivalent to a double lo op space byshowing

g g

that the disjoint union of s is a braiding monoidal category

Lemma The disjoint union of s is a braided monoidal category with the product

inducedbytheF product

Proof Let x y x y b e generators of the fundamental group of S which are induced

g g g

by the Dehn twists a b a b resp ectively The mapping class group can b e iden

g g g

tied with the subgroup of the automorphism group of the free group on x y x y

g g

that consists of the automorphisms xing the fundamental relator R x y x y x y

g g

The binary op eration on q induced by the F pro duct can b e identied with the op

g g

eration taking the free pro duct of the automorphisms The rsbraiding on the free group

on x y x y can b e expressed by

g g

x x

y y

x x

r sr

y y

r sr

x S x S

y S y S

x S x S

r s s

y S y S

r s s

where S x y x y x y

s s s s sr sr

It is easy to see that the rsbraiding xes the fundamental relator R

Moreover the rsbraiding makes the diagrams in d of Denition commute

Lemma explains the pseudo double lo op space structure on the union of the classifying

spaces of the mapping class groups observed by E Miller Lemma and Lemma imply

the following

Theorem The group completion of q B is the homotopy type of a double loop

g g

space

Braid structure

Let B denote Artins braid group B has n generators andisspecied

n n n

by the following presentation

if ji j j

i j j j

for i n

i i i i i i

THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

It has b een observed for manyyears that there are certain connections b etween the braid

groups and the mapping class groups In this section we intro duce a new kind of braid

structure in the mapping class groups s in an explicit form This explicit expression

enables us to deal with a kind of DyerLashof op eration or Browder op eration in an explicit

form It seems p ossible for us to get further applications of the formula of the braid structure

in the future First let us express explicitly the braiding on genus surface is

generated by the Dehn twists a b a b Let x y x y be generators of S

which are induced b y a b a b resp ectively Regard a b a b as automorphisms

on F Then wehave

fx y x y g

a y y x

b x x y

a y y x

b x x y

x x x y x x x y x x

y x x y x x x y x y x y x x

x x y x y x

These automorphisms x the generators that do not app ear in the ab ove list

The braiding in genus should b e expressed in terms of the elements a b a b

and should b e sp ecied on the generators of S by the formulas

x x

y y

x y x x x y

y y x y x y

We need a hard calculation to get such a braiding By using a computer program wecould

get the following explicit formula for the braid structure

Lemma The braiding for the monoidal structure in genus is given by

b a a b a a b a b a a b a

The braid group of all braidings in the mapping class group of genus g is generated by

b a a b a a b a b a a b a

i i i i i i i i i i i i i i i

for i g We can obtain the following formula for the rsbraiding in terms

of the braiding generators

r r r s s r r r s

ZBIGNIEW FIEDOROWICZ AND YONGJIN SONG

or alternatively as

r r r r r s r s s

Remark The braid structure gives rise to the double lo op space structure so it is

supp osed to b e related to the DyerLashof op eration Let D B b e the obvious

g g

map given by

b if i is o dd

if i is even

F Cohen in dealt with this map D He said that the homology homomorphism D

induced by D is trivial b ecause D preserves the DyerLashof op eration Precisely sp eaking

he made a commutative diagram

B B B

p g pg

D B

y y

p g pg

where is the analogue of the DyerLashof op eration it should b e rather Browder op eration

According to his denition B is mapp ed by to b b His denition

of however is not welldened This can b e detected by mapping to Sp Z Here

Sp Z is the automorphism group of H S Z The map Sp Z is describ ed

as follows

C C B B

b a

A A

B B C C

a b

A A

C B

The map sends b b to

Wehave

a a a

THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

This element must commute with a a is mapp ed to

B C C B

How and a is mapp ed to

A A

B C

ever neither of these two matrices commutes with which corresp onds to

The braiding structure plays a key role in the correct formula for which should b e

the following

b a a b a a b a b a a b a

Let C b e the little ncub e op erad Let Y be an nfold lo op space Then Y is a C space

n n

so there is a map

Y C Y

It is known that C has the same homotopytyp e as S Hence the ab ove map induces

a homology op eration

H Y H Y H Y

i j ij n

which is called the Browder op eration It is easy to see that if Y is a C space then the

Browder op eration equals zero

Let X be the group completion of q B Since X is homotopy equivalent to a

g g

space it is up to homotopya C space It is natural to raise the question whether X is

a C space or not The answer is negative In the pro of of the following theorem the braid

formula again plays a key role

Theorem Let X be the group completion of q B The double loop space

g g

structurecannot be extended to the triple loop space structure

Proof Weshow that the Browder op eration

H X H X H X

i j ij

is nonzero for X Wehave the map

C X X

Note that C has the same homotopy typ e as S By restricting the map to each

connected comp onentwe get

S B B B

g g g

ZBIGNIEW FIEDOROWICZ AND YONGJIN SONG

This map is in the group level denoted by the map

g g

which is same as describ ed in Remark In order to showthat is nonzero it suces to

show that

H B H B H B

is nonzero The image of the map equals the image of the homology homomorphism

H S H B induced b y the map S B which is the restriction of

the map S B B B The map sends the generator of H S to the

ab elianization class of

b a a b a a b a b a a b a

which is nonzero since the isomorphism H is natural

References

J S Birman On Siegels modular group Math Ann

and B Wa jnryb Errata Presentations of the mapping class group Israel Journal of Math

F Cohen Homology of mapping class groups for surfaces of low genusContemp Math Part I I

Z Fiedorowicz The symmetric bar construction preprint

A Joyal and R Street Braided tensor categories Macquire Math Rep orts March

R KirbyandP Melvin The manifold invariants of Witten and ReshetikhinTuraev for sl C Invent

Math

S MacLane Categorical algebra Bull AMS

J S Maginnis Braids and mapping class groups PhD thesis Stanford University

J PMay The geometry of iteratedloop spaces Lec Notes in Math SpringerVerlag

E Miller The homology of the mapping class group J Dierential Geometry

N Reshetikhin and V G Turaev Invariants of manifolds via link polynomials and quantum groups

Invent Math

Ribbon graph and their invariants derived from quantum groups Comm Math Phys

J D Stashe Homotopy associativity of H spaces I Trans AMS

B Wajnryb A simple representation for mapping class group of an orientable surface Israel Journal

of Math

E Witten Quantum eld theory and the Jones polynomial Comm Math Phys

G Wright The ReshetikhinTuraev representation of the mapping class group Preprint

THE BRAID STRUCTURE OF MAPPING CLASS GROUPS

Department of Mathematics

Ohio State University

Department of Mathematics

Inha University

CONTROLLED TOPOLOGICAL EQUIVALENCE OF MAPS

IN THE THEORY OF

STRATIFIED SPACES AND APPROXIMATE FIBRATIONS

Bruce Hughes

Abstract Ideas from the theory of top ological stability of smo oth maps are trans

p orted into the controlled top ological category For example the controlled top ologi

cal equivalence of maps is discussed These notions are related to the classication of

manifold approximate brations and manifold stratied approximate brations In

turn these maps form a bundle theory which can b e used to describ e neighborhoods

of strata in top ologically stratied spaces

Introduction

We explore some connections among the theories of top ological stability of maps

controlled top ology and stratied spaces The notions of top ological equivalence

of maps and lo cally trivial families of maps play an imp ortant role in the theory of

top ological stability of smo oth maps Weformulate the analogues of these notions in

the controlled top ological category for two reasons First the notion of controlled

top ological equivalence of maps is a starting point for formulating a top ological

version of Mathers theory of the top ological stability of smo oth maps Recall that

Mather proved that the top ologically stable maps are generic for the space of all

smo oth maps with the C top ology b etween closed smo oth manifolds see Mather

Gibson Wirthmuller du Plessis and Lo oijenga The hop e is to identify an

analogous generic class for the space of all maps with the compactop en top ology

between closed top ological manifolds Controlled top ology at least gives a place to

b egin sp eculations Second the controlled analogue of lo cal triviality for families

of maps is directly related to the classication of approximate brations b etween

manifolds due to Hughes Taylor and Williams We elucidate that relation

in x

Another imp ortant topic in the theory of top ological stability of smo oth maps is

that of smo othly stratied spaces cf Mather Quinn initiated the study

of top ologically stratied spaces and Hughes has shown that manifold

Mathematics Subject Classication Primary N N A Secondary R

R N

Key words and phrases stratied space approximate bration controlled top ology top ologi

cal stability of maps strata neighb orho o d germ Thoms isotopy lemmas

Supp orted in part by NSF Grant DMS and by the Vanderbilt University Research

Council

Presented at the Symp osium on Surgery and Geometric Top ology Josai UniversitySakado

Japan Septemb er

BRUCE HUGHES

stratied approximate brations form the correct bundle theory for those spaces

The classication of manifold approximate brations via controlled top ology men

tioned ab ove extends to manifold stratied approximate brations hence wehave

another connection b etween controlled top ology and stratied spaces This classi

cation of manifold stratied approximate brations is the main new result of this

pap er

Two essential to ols in stability theory are Thoms two isotopy lemmas In

xweformulate an analogue of the rst of these lemmas for top ologically stratied

spaces A nonprop er version is also stated

It should be noted that in his address to the International Congress in

Quinn predicted that controlled top ology would have applications to the top olog

ical stability of smo oth maps In particular controlled top ology should be

applicable to the problem of characterizing the top ologically stable maps among

all smo oth maps More recently Capp ell and Shaneson suggested that top o

logically stratied spaces should play a role in the study of the lo cal and global

top ological typ e of top ologically smo oth maps the connection is via the mapping

cylinder of the smo oth map While the sp eculations in this pap er are related to

these suggestions they dier in that it is suggested here that controlled top ology

might b e used to study a generic class of top ological rather than smo oth maps

Topological equivalence and locally trivial families of maps

We recall some denitions from the theory of top ological stability of smo oth

maps see Damon du Plessis and Wall Gibson Wirthmuller du Plessis and

Lo oijenga Mather

Denition Twomapsp X Y p X Y are topological ly equivalent

if there exist homeomorphisms h X X and g Y Y such that p h gp

so that there is a commuting diagram

X X

p p

y y

Y Y

Denition A smo oth map p M N between smo oth manifolds is topo

logical ly stable if there exists a neighborhood V of p in the space of all smo oth

maps C M N such that for all p V p is top ologically equivalenttop

The space C M N is given the Whitney C top ology Thom conjectured

ved that the top ologically stable maps are generic in C M N in and Mather pro

fact they form an op en dense subset see The pro of yields a stronger

result namely that the strongly top ologically stable maps are dense see

Denition A smo oth map p M N between smo oth manifolds is strongly

topological ly stable if there exists a neighborhood V of p in C M N such that

for all p V there exists a topological ly trivial smo oth oneparameter family

p M I N joining p to p This means there exist continuous families

CONTROLLED TOPOLOGICAL EQUIVALENCE

fh M M j t g and fg N N j t g of homeomorphisms such

t t

that p g p h for all t I so that there is a commuting diagram

t t

M M

p p

y y

N N

The notion of trivialityforthe oneparameter family of maps in the denition

ab ove can b e generalized to arbitrary families of maps Wenow recall that denition

and the related notion of lo cal triviality cf

Denition Consider a commuting diagram of spaces and maps

E E

p p

y y

B B

f is trivial over B if there exist spaces F and F a map q F F and

homeomorphisms h E F B g E F B such that the following

diagram commutes

p f p

E B E B

id id

B B

y y y y

pro j pro j q id

F B B B F B

f is local ly trivial over B if for every x B there exists an op en neighb or

ho o d U of x in B suchthatf j p U p U is trivial over U

is the model of the family f In either case q F F

Remarks

The mo del q F F is welldened up to top ological equivalence

Both p E B and p E B are bre bundle pro jections with bre

F and F resp ectively

x is top ologically equivalentto x p For every x B f f j p

q F F

One step in Mathers pro of that the top ologically stable smo oth maps form

an op en dense subset is to show that certain families of maps are lo cally

trivial Thoms second isotopy lemma is used for this

A bre preserving map is a map which preserves the bres of maps to a given

space usually a k simplex or an arbitrary space B Sp ecicallyif X B and

Y B are maps then a map f X Y is bre preserving over B if f

There is a notion of equivalence for families of maps over B

BRUCE HUGHES

Denition

Two lo cally trivial families of maps over B

E E

and

p p

y y

B B

are topological ly equivalent provided there exist homeomorphisms

h E E and h E E

which are bre preserving over B and f h h f that is the following

diagram commutes

p f p

B E E B

id h h id

B B

y y y y

p p

B E E B

Let A q B denote the set of top ological equivalence classes of lo cally

trivial families of maps over B with mo del q F F

The set A q B can be interpreted as a set of equivalence classes of certain

bre bundles over B as follows Let TOPq b e the top ological group given bythe

pullback diagram

TOPq TOPF

y y

MapF TOPF F

where q hq h and q g g q That is

TOPq fh g TOPF TOPF j qh gqg

Note that TOPq is naturally a subgroup of TOPF q F viah g h q g Let

A q B denote the set of bundle equivalence classes of bre bundles over B with

bre F q F and structure group TOPq

Prop osition There is a bijection A q B A q B In particular if

B is a separable metric space then there is a bijection A q B B BTOPq

The function is dened by sending a lo cally trivial family

E E

p p

y y

B B

to the bre bundle p q p E q E B whose total space is the disjoint union of

E and E The fact that is a bijection is fairly straightforward to prove Atany

rateitfollows from a more general result in x see Theorem and the comments

follo wing it

CONTROLLED TOPOLOGICAL EQUIVALENCE

Controlled topological equivalence

We prop ose a denition of top ological equivalence in the setting of controlled

top ology and use it to make some sp eculations ab out generic maps b etween top o

logical manifolds

The mapping cylinder of a map p X Y is the space

cylpX I q Y fx px j x X g

There is a natural map cylp I dened by

x t t if x t X I

y if y Y

I If p X Y For clarication the map will sometimes b e denoted cylp

cylp I are the and p X Y are maps and cylp I and

p p

h natural maps then a homeomorphism h cylp cylp is level if

p p

level

Let TOP p denote the simplicial group of level homeomorphisms from cylp

level

onto itself That is a k simplex of TOP p consists of a parameter family

k k

of level homeomorphisms h cylp cylp The group TOPp as

dened in the previous section has a simplicial version the singular complex of

level

the top ological group and as such is a simplicial subgroup of TOP p For

example a pair of homeomorphisms h X X g Y Y suchthat ph gp

induces a level homeomorphism

x t hxt if x X

cylp cylp

y Y y g y if

Denition Two maps p X Y p X Y are control ledtopological ly

equivalent if there exists a level homeomorphism h cy l p cylp

Note that a level homeomorphism h cylp cylp induces by restriction

a oneparameter family h X X t of homeomorphisms and a

homeomorphism g Y Y If all the spaces involved are compact metric then

gp lim p h

and such data is equivalenttohaving a level homeomorphism cf

This formulation should be compared with the formulation of top ological

equivalence in Denition

Denition Two maps p X Y p X Y between compact metric

spaces are weakly control led topological ly equivalent if there exist continuous families

fh X X j tg and fg Y Y j tg of homeomorphisms

t t

lim g p h suchthat p

t t

The limit ab ove is taken in the sup metric which is the metric for the compact

op en top ology The space C X Y of maps from X to Y is given the compactop en

top ology

BRUCE HUGHES

Denition A map p X Y between compact metric spaces is weakly

control led topological ly stable if there exists a neighborhood V of p in C X Y

such that for all p V and there exists a map p X Y suchthat p is

weakly controlled top ologically equivalentto p and p is close to p

Many of the results in the theory of singularities have a mixture of smo oth and

top ological hyp otheses and conclusions This is the case in Mathers theorem on

the genericness of top ologically stable maps among smo oth maps One direction

that controlled top ology is likelytotake is in nding the top ological underpinnings

in this area The following sp eculation is meant to b e a step towards formulating

what might b e true

Sp eculation If M and N are closed topological manifolds then the weakly

control ledtopological ly stable maps from M to N are generic in C M N

This might b e established byshowing that the stratied systems of approximate

brations are dense and also weakly controlled top ologically stable see Hughes

and Quinn for stratied systems of approximate brations As evidence for this

line of reasoning note that Chapmans work shows that manifold approximate

brations are weakly controlled top ologically stable

Another line of sp eculation concerns p olynomial maps b etween euclidean spaces

It is known that the classication of p olynomial maps up to smo oth equivalence

diers from their classication up to top ological equivalence cf Thom Fakuda

Nakai What can b e said ab out the classication of p olynomial maps up

to controlled top ological equivalence

Controlled locally trivial families of maps

Analogues in controlled top ology of lo cally trivial families of maps are dened

In fact we dene a mo duli space of all such families

Denition Consider a commuting diagram of spaces and maps

E E

p p

y y

B B

f is control led trivial over B if there exist spaces F and F amapq F

F and a homeomorphism H cylf cylq B such that the following

diagram commutes

f c

I B cylf

id id

I B

y y y

pro j

I B cylq B

where c cylf B is given by

cx t p xp f x if x t E I

cy p y if y E

CONTROLLED TOPOLOGICAL EQUIVALENCE

pro j

and is the comp osition cylq B cylq I

f is control ledlocal ly trivial over B if for every x B there exists an op en

neighb orho o d U of x in B such that f j p U p U is controlled

trivial over U

In either case q F F is the model of the family f

Remarks

The mo del q F F is welldened up to controlled top ological equiva

lence

Both p E B and p E B are bre bundle pro jections with bre

F and F resp ectively

For every x B f f j p x p x is controlled top ologically

equivalentto q F F

There is a notion of controlled equivalence for families of maps over B

Denition

Two controlled lo cally trivial families of maps over B

E E

and

p p

y y

B B

are control led topological ly equivalent provided there exists a level homeo

morphism

H cylf cylf

which is bre preserving over B in the sense that the following diagram

commutes

cylf cylf

y y

B B

where c is given by

cx t p f xp x if x t E I

cy p y if y E

and c is given by

c x t p f xp x if x t E I

c y p y if y E

Let B q B denote the set of controlled top ological equivalence classes of

lo cally trivial families of maps over B with mo del q F F

BRUCE HUGHES

In the next section wewillshow that the set B q B canbeinterpretedasaset

of equivalence classes of certain bre bundles over B in analogy with Prop osition

see Theorem But rst wewilldene the mo duli space of all controlled

lo cally trivial families of maps over B with mo del q F F This is done in the

setting of simplicial sets as follows

Dene a simplicial set B q B so that a typical k simplex of B q B consists

of a commuting diagram

E E

p p

y y

k k

B B

which is a controlled lo cally trivial family of maps over B with mo del q

F F Thusavertex of B q B is a controlled lo cally trivial family of maps

over B with mo del q F F As in we also need to require that these

spaces are reasonably emb edded in an ambient universe but we will ignore that

technicality in this pap er Face and degeneracy op erations are induced from those

on the standard simplexes As in this simplicial set satises the Kan condition

Denition The mapping cylinder construction takes a controlled lo cally

trivial family of maps

E E

p p

y y

B B

to the mapping cylinder cylf together with the natural map f cylf B

Note that the controlled lo cally trivial condition on f means that f cylf

level

B is a bre bundle with bre cylq and structure group TOP q where q is the

mo del of f If

E E

p p

y y

B B

is another controlled lo cally trivial family of maps over B with mo del q then to

havea controlled top ological equivalence H cylf cylf as in Denition

means precisely to have a bundle isomorphism from f tof

Prop osition There is a bijection B q B B q B

Proof In order to see that the natural function B q B B q B is well

dened supp ose

E E

p p

y y

B B

CONTROLLED TOPOLOGICAL EQUIVALENCE

is a lo cally trivial family of maps with mo del q F F Then by the remarks

ab ove f cylf B is a bre bundle with bre cylq and structure group

level

TOP q Thus there is a bundle isomorphism from the restriction of f over

B fg to the restriction of f over B fg and the remarks ab ove further show

that this isomorphism gives a controlled top ological equivalence from

f j f j

p B fg p B fg p B fg p B fg

p j p j p j p j

y y y y

id id

B B

B fg B fg B fg B fg

showing that the function is welldened The function is obviously surjective so

it remains to see that it is injective To this end supp ose that

E E

and

p p

y y

B B

are controlled top ologically equivalent with a level homeomorphism H cylf

cylf as in Denition Let h E E and h E E b e the restrictions

of H to the top and b ottom of the mapping cylinders resp ectively Then there is

an induced commutative diagram

cylh cyl h

y y

B B

which is a simplex in B q B from f to f

Bundles with mapping cylinder fibres

In this section we show that controlled lo cally trivial families of maps over B

can b e interpreted as bre bundles over B with bre the mapping cylinder of the

mo del Reduced structure groups are discussed as well as a relative situation in

which the target bundle over B is xed

Let B b e a xed separable metric space Let B q B denote the set of bundle

with bre cylq and structure group equivalence classes of bre bundles over B

level

TOP q Dene B q B to be the simplicial set whose k simplices are bre

level

bundles over B with bre cylq and structure group TOP q The fol

lowing result is wellknown cf

Prop osition Thereare bijections

level

B q B B q B B BTOP q

The mapping cylinder construction of Denition has the following simplicial

version

BRUCE HUGHES

Denition The mapping cylinder construction is the simplicial map

B q B B q B

dened by sending a diagram

E E

p p

y y

k k

B B

to cylf B Note that the lo cal triviality condition on f implies that

cylf B is a bre bundle pro jection with bre cylq and structure group

level

TOP q

The rst part of the following result is pro ved in The second part follows

from the rst part together with Prop ositions and

Theorem The mapping cylinder construction denes a homotopy equivalence

level

q B q B B q B In particular B q B B q B B BTOP

level

Reduced structure groups Let G b e a simplicial subgroup of TOP q We

will now generalize the discussion ab ove to the situation where the structure group

is reduced to G

Denition Consider a controlled lo cally trivial family

E E

p p

y y

B B

F Then f is Glocal ly trivial over B provided there exists with mo del q F

an op en cover U of B suchthat f is controlled trivial over U for each U U via a

trivializing homeomorphism

H cylf j p U p U cylq B

These trivializing homeomorphisms are required to have the prop erty that if U V

U and x U V then

H H j cylq fxgcylq fxg

tofG is an elemen

Let B q B G b e the simplicial set whose k simplices are the Glo cally trivial

families of maps over B with mo del q F F For example

level

B q B TOP q B q B

CONTROLLED TOPOLOGICAL EQUIVALENCE

Denition can b e extended in the obvious way to dene what it means for

two Glo cally trivial families to b e Gcontrol ledtopological ly equivalent the home

omorphism H is required to b e a family of homeomorphisms in the group G and

B q B G denotes the set of equivalence classes In analogy with Prop osition

there is a bijection

B q B G B q B G

Likewise B q B G denotes the set of bundle equivalence classes of bre bundles

over B with bre cylq and structure group G and B q B G is the simplicial

set whose k simplices are bre bundles over B with bre cylq and structure

group G In analogy with Prop osition there are bijections

B q B G B q B G B BG

Moreover the pro of of Theorem can be seen to give a pro of of the following

result cf x

Theorem The mapping cylinder construction denes a homotopy equivalence

B q B G B q B G In particular B q B G B q B G B BG

As an example consider the group TOPq of x It was pointed out at the

level

is naturally a subgroup of TOP q Note that b eginning of x that TOPq

B q B TOPq A q B andB q B TOPq A q B so that Prop osition

follows directly from Theorem

Fixed target bundle There are also relativeversions of the preceding results in

which the bundle p E B is xed For example B q rel p E B isthe

set of controlled lo cally trivial families of maps of the form

E E

p p

y y

B B

Two such families f E E and f E E are control led topological ly

equivalent rel p if the homeomorphism H cylf cylf of Denition is

required to b e the identityonE There are analogous denitions of the following

B q rel p E B

Denition The group of control led homeomorphisms of q is the subgroup

c level

TOP q ofTOP q consisting of all level homeomorphisms h cylq

k k

cylq such that hjF id k

Note that TOP q isthekernel of the restriction homomorphism

level

TOP q TOPF

BRUCE HUGHES

Let pb B BTOPF be the classifying map for the bundle p Thus

B q rel p E B is in onetoone corresp ondence with the set of vertical

level

homotopy classes of lifts of pb B BTOPF toBTOP q BTOPF

level

BTOP q

B BTOPF

The following result follows from the pro ofs of the preceding results

Prop osition

B q rel p E B B q rel p E B

the mapping cylinder construction denes a homotopy equivalence

B q rel p E B B q rel p E B

Reduced structure group and xed target bundle There are versions of

these relative results when the structure groups are reduced to G as b efore The

sets and simplicial sets involved are denoted as follows

B q G rel p E B

The following result records the analogous bijections and homotopyequivalences

Prop osition

B q G rel p E B B q G rel p E B

the mapping cylinder construction denes a homotopy equivalence

B q G rel p E B B q G rel p E B

Manifold stratified spaces

There are many naturally o ccurring spaces which are not manifolds but which

are comp osed of manifold pieces those pieces b eing called the strata of the space

Examples include p olyhedra algebraic varieties orbit spaces of many group actions

on manifolds and mapping cylinders of maps b etween manifolds Quinn has

intro duced a class of stratied spaces called by him manifold homotopically strat

study of purely top ological ied sets with the ob jective to give a setting for the

stratied phenomena as opp osed to the smo oth and piecewise linear phenomena previously studied

CONTROLLED TOPOLOGICAL EQUIVALENCE

Roughly the stratied spaces of Quinn are spaces X together with a nite l

tration by closed subsets

m m

X X X X X

i i

such that the strata X X n X are manifolds with neighborhoods in X X

i i k

for k i which have the lo cal homotopy prop erties of mapping cylinders of

brations These spaces include the smo othly stratied spaces of Whitney

Thom and Mather for historical remarks on smo othly stratied spaces see

Goresky and MacPherson as well as the lo cally conelike stratied spaces of

Sieb enmann and hence orbit spaces of nite groups acting lo cally linearly on

manifolds

Capp ell and Shaneson have shown that mapping cylinders of smo othly strat

ied maps betw een smo othly stratied spaces are in this class of top ologically

stratied spaces even though it is known that such mapping cylinders need not b e

smo othly stratied see and Hence the stratied spaces of Quinn arise

naturally in the category of smo othly stratied spaces For a comprehensivesurvey

of the classication and applications of stratied spaces see Weinb erger

Smo othly stratied spaces have the prop erty that strata have neighb orho o ds

which are mapping cylinders of bre bundles a fact which is often used in arguments

involving induction on the number of strata Such neighb orho o ds fail to exist in

general for Sieb enmanns lo cally conelike stratied spaces For example it is known

that a top ologically lo cally at submanifold of a top ological manifold which is

an example of a lo cally conelike stratied space with two strata may fail to have

a tubular neighborhood see Rourke and Sanderson However Edw ards

proved that such submanifolds do have neighb orho o ds which are mapping cylinders

of manifold approximate brations see also On the other hand examples

of Quinn and Steinb erger and West show that strata in orbit spaces of

nite groups acting lo cally linearly on manifolds mayfailtohave mapping cylinder

neighborhoods In Quinns general setting mapping cylinder neighb orho o ds may

fail to exist even lo cally

The main result announced in and restated here in x gives an eective

substitute for neighb orho o ds which are mapping cylinders of bundles Instead of

bre bundles we use manifold stratied approximate brations and instead of

mapping cylinders we use teardrops This result should b e thought of as a tubular

neighb orho o d theorem for strata in manifold stratied spaces

We now recall the concepts needed to precisely dene the manifold stratied

spaces of interest see A subset Y X is forward tame in X

if there exist a neighborhood U of Y in X and a homotopy h U I X such

that h inclusion U X h jY inclusion Y X for each t I h U Y

and hU n Y X n Y

Dene the homotopy link of Y in X by

holinkX Y f X j t Y i t g

Evaluation at denes a map q holinkX Y Y called holink evaluation

m m

Let X X X X X b e a space with a nite ltration

i i i

by closed subsets Then X is the iskeleton and the dierence X X n X is

called the istratum

BRUCE HUGHES

AsubsetA of a ltered space X is called a pure subset if A is closed and a union

of comp onents of strata of X For example the skeleta are pure subsets

The stratied homotopy link of Y in X denoted holink X Y consists of all

in holinkX Y suchthat lies in a single stratum of X The stratied

homotopy link has a natural ltration with iskeleton

i i

holink X Y f holink X Y j X g

s s

The holink evaluation at restricts to a map q holink X Y Y

If X is a ltered space then a map f Z A X is stratum preserving along

A if for each z Z f fz g A lies in a single stratum of X In particular a map

f Z I X is a stratum preserving homotopy if f is stratum preserving along I

Denition A ltered space X is a manifold stratied space if the following

four conditions are satised

Manifold strata X is a lo cally compact separable metric space and each

stratum X is a top ological manifold without b oundary

Forward tameness For each k i the stratum X is forward tame in

X X

i k

Normal brations For each ki the holink evaluation q holinkX

X X X is a bration

k i i

Finite domination For each i there exists a closed subset K of the

stratied homotopy link holink X X such that the holink evaluation map

K X is prop er together with a stratum preserving homotopy

i i

h holink X X I holink X X

s s

which is also bre preserving over X ie qh q for each t I such that

h id and h holink X X K

Manifold stratified approximate fibrations

The denition of an approximate bration as given in was generalized in

m n

to the stratied setting Let X X X and Y Y Y

be ltered spaces and let p X Y be a map p is not assumed to be stratum

preserving Then p is said to b e a stratied approximate bration provided given

anyspaceZ and any commuting diagram

Z X

y y

Z I Y

where F is a stratum preserving homotopy there exists a stratied control led so

lution ie a map F Z I X which is stratum preserving along

I such that F z tf z for eachz t Z and the function

F Z I Y dened by F jZ I pF and F jZ I fg F id

is continuous

A stratied approximate bration b etween manifold stratied spaces is a mani

oximate bration if in addition it is a prop er map ie inverse fold stratied appr images of compact sets are compact

CONTROLLED TOPOLOGICAL EQUIVALENCE

Teardrop neighborhoods

Given spaces X Y and a map p X Y R the teardrop of p see is the

space denoted by X Y whose underlying set is the disjoint union X q Y with

the minimal top ology such that

X X Y is an op en emb edding and

the function c X Y Y dened by

px if x X

x if x Y

is continuous

The map c is called the tubular map of the teardrop or the teardrop col lapse The

tubular map terminology comes from the smo othly stratied case see

This is a generalization of the construction of the op en mapping cylinder of

amap g X Y Namely cylg is the teardrop X R Y

g id

Theorem If X and Y are manifold stratiedspaces and p X Y R is a

manifold stratied approximate bration then X Y is a manifold stratiedspace

with Y apure subset

In this statement Y R and X Y are given the natural stratications

The next result from is a kind of converse to this prop osition First some

more denitions A subset Y of a space X has a teardrop neighborhood if there exist

a neighborhood U of Y in X and a map p U n Y Y R such that the natural

function U n Y Y U is a homeomorphism In this case U is the teardrop

neighborhood and p is the restriction of the tubular map

Theorem Teardrop Neighborhood Existence Let X be a manifold

stratied space such that al l components of strata have dimension greater than

and let Y beapure subset Then Y has a teardrop neighborhood whose tubular map

c U Y

is a manifold stratiedapproximate bration

A complete pro of of this result will b e given in but sp ecial cases are in

and

The next result from concerns the classication of neighb orho o ds of pure

subsets of a manifold stratied space Given a manifold stratied space Y astrat

ied neighborhood of Y consists of a manifold stratied space containing Y as a

pure subset Two stratied neighb orho o ds X X of Y are equivalent if there exist

neighborhoods U U of Y in X X resp ectively and a stratum preserving home

h U U such that hjY id A neighborhood germ of Y is an omorphism

equivalence class of stratied neighborhoods of Y

Theorem Neighborhood Germ Classication Let Y be a manifold

stratied space such that al l components of strata have dimension greater than

Then the teardrop construction induces a onetoone correspondence from con

trol led stratum preserving homeomorphism classes of manifold stratied approxi

mate brations over Y R to neighborhood germs of Y

BRUCE HUGHES

Applications of Teardrop Neighborhoods

Teardrop neighb orho o ds can also be used in conjunction with the geometric

theory of manifold approximate brations to study the geometric top ology of

manifold stratied pairs Examples of results proved using teardrop technology are

stated in this section Details will app ear in

Theorem Parametrized Isotopy Extension Let X be a manifold strati

edspace such that al l components of strata have dimension greater than letY be

k k

apure subset of X letU be a neighborhoodofY in X and let h Y Y

beak parameter stratum preserving isotopy Then thereexistsak parameter iso

k k k

topy h X X extending h and supportedonU

This generalizes results of Edwards and Kirby Sieb enmann and Quinn

The next result is a top ological analogue of Thoms First Isotopy Theorem

and can b e viewed as a rst step towards a top ological theory of top ological stability

Theorem First Top ological Isotopy Let X be a manifold stratiedspace

and let p X R be a map such that

i p is proper

ii for each stratum X of X pj X R is a topological submersion

i i

iii for each t R the ltration of X restricts to a ltration of p t giving

p t the structure of a manifold stratie dspace such that al l components

of strata have dimension greater than

Then p is a bund le and can be trivializedbyastratum preserving homeomorphism

that is there exists a stratum preserving homeomorphism h p R X

such that ph is projection

Here is a nonprop er version of the preceding result

Theorem Nonprop er First Top ological Isotopy Let X be a manifold

stratiedspace and let p X R be a map such that

i if X is a proper map and p p X R then

R is a manifold stratiedspace the teardrop X

ii for each stratum X of X pj X R is a topological submersion

i i

iii for each t R the ltration of X restricts to a ltration of p t giving

p t the structure of a manifold stratiedspace such that al l components

of strata have dimension greater than

Then p is a bund le and can be trivializedbyastratum preserving homeomorphism

that is there exists a stratum preserving homeomorphism h p R X

such that ph is projection

Classifying manifold stratified approximate fibrations

Some applications of teardrop neighb orho o ds are combined with the material in

xon bundles with mapping cylinder bres in order to presentaclassication of

manifold stratied approximate brations at least when the range is a manifold

generalizing the classication of manifold approximate brations in and

CONTROLLED TOPOLOGICAL EQUIVALENCE

For notation let B b e a connected imanifold without b oundary and let q V

R be a manifold stratied approximate bration where all comp onents of strata

of V have dimension greater than A stratied manifold approximate bration

p X B has bre germ q if there exists an emb edding R B such that pj

i i

p R R is controlled stratum preserving homeomorphic to q that is there

i i

exists a stratum preserving level homeomorphism cylq cylpj p R R

where the mapping cylinders have the natural stratications

The following result shows that bre germs are essentially unique For nota

i i

tion let r R R be the orientation reversing homeomorphism dened by

r x x x x x x

i i

Theorem Let p X B be a manifold stratiedapproximate bration such

that al l components of strata have dimension greater than Let g R B k

i i

be two open embeddings Then pj p g R g R is control led stratum

i i i

preserving homeomorphic to either pj p g R g R or pj p g R

rg R

Proof The pro of follows that of the corresp onding result for manifold approxi

mate brations in Cor The stratied analogues of the straightening

phenomena are consequences of the teardrop neighb orho o d results The

use of Sieb enmanns Technical Bundle Theorem is replaced with the nonprop er

top ological version of Thoms First Isotopy Lemma in x

There is a mo duli space MSAFB of all manifold stratied approximate

brations over B with bre germ q It is dened as a simplicial set with a typ

k k

ical k simplex given by a map p X B such that for each t

pj p t B ftg is a manifold stratied approximate bration with bre germ

q and there exists a stratum preserving homeomorphism p X whichis

bre preserving over There is also a technical condition giving an emb edding

in an ambientuniverse cf

The pro of of the next prop osition follows that of the corresp onding result for

manifold approximate brations in The necessary stratied versions of the

manifold approximate bration to ols are in and and follow from teardrop

technology

Prop osition MSAF B is in onetoone correspondence with the set of

control led stratum preserving homeomorphism classes of stratied manifold approx

imate brations over B with bre germ q

level

Let TOP q denote the simplicial group of self homeomorphisms of the map

ping cylinder cylp which preserve the mapping cylinder levels and are stratum

preserving with resp ect to the induced stratication of cylq Note that there is a

level

q TOP restriction homomorphism TOP

Let B B denote the top ological tangent bundle of B Consider B as an

op en neighb orho o d of the diagonal in B B so that B B is rst co ordinate

level

q relB B pro jection As in xwe can form the simplicial set B q TOP

level

whichwe denote simply by B q TOP q rel B

The dierential

level

d MSAF B B q TOP q rel B

q s

BRUCE HUGHES

is a simplicial map whose denition is illustrated on vertices as follows for higher

dimensional simplices the construction is analogous cf If p X B is a

vertex of MSAF B then form

id p B X B B

and let

p pj E p B B

Thus there is a commuting diagram

E B

y y

B B

It follows from the stratied straightening phenomena that the lo cal triviality

condition is satised so that the diagram is a vertex of

level

B q TOP q relB

Once again the pro of of the following result follows that of the corresp onding man

ifold approximate bration result in using the stratied results of and

Theorem MSAF Classication The dierential

level

q relB d MSAF B B q TOP

is a homotopy equivalence

Corollary Control led stratum preserving homeomorphism classes of strat

ied manifold approximate brations over B with bre germ q are in onetoone

correspondence with homotopy classes of lifts of the map B BTOP which

level

classies the tangent bund le of B toBTOP q

level

BTOP q

B BTOP

Proof Combine Theorem Prop osition and Prop osition

Finally observe that Corollary can b e combined with Theorem to give

a classication of neighborhood germs of B with xed lo cal typ e

CONTROLLED TOPOLOGICAL EQUIVALENCE

References

S Capp ell and J Shaneson The mapping cone and cylinder of a stratiedmap Prosp ects in

top ology Pro ceedings of a conference in honor of William Browder F Quinn ed Ann of

Math Studies vol Princeton Univ Press Princeton pp

T A Chapman Approximation results in topological manifolds Mem Amer Math So c

J Damon Applications of singularity theory to the solutions of nonlinear equationsTop o

logical Nonlinear Analysis Degree SingularityandVariations M Matzeu and A Vignoli

eds Progress in Nonlinear Dierential Equations and Their Applications vol Birkhauser

Boston pp

K H Dovermann and R Schultz Equivariant surgery theories and their periodicity properties

Lecture Notes in Math SpringerVerlag New York

A du Plessis and T Wall The geometry of topological stability London Math So c Mono

New Series Oxford Univ Press Oxford

R D Edwards TOP regular neighborhoods handwritten manuscript

R D Edwards and R C Kirby Deformations of spaces of imbeddings Ann of Math

T Fakuda Typ es topologiques des polynomes Publ Math Inst Hautes Etudes Sci

C G Gibson K Wirthmuller A A du Plessis and E J N Lo oijenga Topological stability

of smooth mappings Lecture Notes in Math SpringerVerlag New York

M Goresky and R MacPherson Stratied Morse theory Ergeb Math Grenzgeb

SpringerVerlag New York

B Hughes Approximate brations on topological manifoldsMichigan Math J

Geometric topology of stratied spaces ERAAMS to app ear http

wwwamsorgjournalsera

The geometric topology of stratiedspaces in preparation

Stratiedpath spaces and brations in preparation

B Hughes and A Ranicki Ends of complexesCambridge Tracts in Math Cambridge

Univ Press Cambridge

B Hughes L Taylor S Weinb erger and B Williams Neighborhoods in stratiedspaces with

two strata preprint

B Hughes L Taylor and B Williams Bund le theories for topological manifoldsTrans Amer

Math So c

Manifold approximate brations areapproximately bund lesForum Math

Bounded homeomorphisms over Hadamard manifolds Math Scand

Rigidity of brations over nonpositively curved manifoldsTop ology

J Mather Notes on topological stability Harvard Univ Cambridge photo copied

Stratications and mappings Dynamical Systems M M Peixoto ed Academic

Press New York pp

I Nakai On topological types of polynomial mappingsTop ology

F Quinn Ends of maps II Invent Math

Applications of topology with control Pro ceedings of the International Congress of

Mathematicians Berkeley Amer Math So c Providence RI pp

Homotopical ly stratied sets J Amer Math So c

Speculations on Gromov convergenceofstratiedsetsandRiemannian volume col

lapse Prosp ects in top ology Pro ceedings of a conference in honor of William Browder F

Quinn ed Ann of Math Studies vol Princeton Univ Press Princeton pp

C Rourke and B Sanderson An embedding without a normal bund leInvent Math

BRUCE HUGHES

L Sieb enmann Deformations of homeomorphisms on stratied sets Comment Math Helv

M Steinb erger and J West Equivariant hcobordisms and niteness obstructions Bull

Amer Math So c NS

R Thom Ensembles et morphismes straties Bull Amer Math So c

La stabilite des applications polynomials LEnseignement Mathematique I I

A Verona Stratied mappings structure and triangulability Lecture Notes in Math

SpringerVerlag New York

S Weinb erger The topological classication of stratiedspaces Chicago Lectures in Math

Univ Chicago Press Chicago

H Whitney Local properties of analytic varieties Dierentiable and combinatorial top ology

S Cairns ed Princeton Univ Press Princeton pp

DepartmentofMathematics Vanderbilt University Nashville TN

Email address hughesmathvanderbiltedu

THE ASYMPTOTIC METHOD IN THE

NOVIKOV CONJECTURE

TKato

The famous Hirzebruch signature theorem asserts that the signature of a closed

oriented manifold is equal to the integral of the so called Lgenus An immediate

corollary of this is the homotopyinvariance of LM M The Lgenus is a

characteristic class of tangent bundles so the ab ove remark is a nontrivial fact

The problem of higher signatures is a generalization of the ab ove consideration

Namely we investigate whether the higher signatures are homotopy invariants or

not The problem is called the Novikov conjecture The characteristic numb ers are

closely related to the fundamental groups of manifolds

There are at least two pro ofs of the signature theorem One is to use the cob or

ahSinger index theorem Recall that the dism ring The other is to use the Atiy

signature is equal to the index of the signature op erators The higher signatures

are formulated as homotopyinvariants of b ordism groups of B The problem was

solved using the AtiyahSinger index theorem in many partial solutions Here we

have the indextheoretic approach in mind when considering the higher signatures

Roughly sp eaking a higher signature is an index for a signature op erator with

some co ecients Tointerpret the numb er as a generalized signature one consid

ers homology groups with rational group ring co ecients By doing surgery on the

homology groups we obtain non degenerate symmetric form L over the

group ring It is called the MishchenkoRanicki symmetric signature This element

is a homotopyinvariant of manifolds Mishchenkointro duced Fredholm represen

tations obtaining a number F from a Fredholm representation F and On

the other hand one can construct a virtual bundle over K from a Fredholm

representation By pulling back the bundle through maps from the base mani

folds to K we can make a signature op erator with co ecients Mishchenko

discovered the generalized signature theorem which asserts the coincidence of the

index of the op erators and F Thus a higher signature coming from a Fredholm

representation is an oriented homotopyinvariant

In CGM the authors showed that all higher signatures come from Fredholm

representations for large class of discrete groups including word hyp erb olic groups

They formulated the notion of a prop er Lipschitz cohomology class in group coho

mology It corresp onds to a Fredholm representation in K theory In fact for many

group cohomology can be represented by a prop er discrete groups any class of

Lipschitz cohomology class Their metho d dep ends on the existence of nite di

mensional spaces of Q typ e K

On the other hand for larger classes of discrete groups we cannot exp ect existence

of such go o d spaces In G Gromovintro duced a very large class of discrete groups

TKATO

the quasi geo desic bicombing groups This class is characterized byconvexityofthe

Cayley graph Hyp erb olic groups are contained in the class For the class we cannot

ensure the existence of good spaces as ab ove Moreover it is unknown whether

H Q is zero for suciently large n Toovercome this diculty the following is

shown in K We realize K by innite dimensional space and approximate it

by a family of nite dimensional spaces By applying the metho d of CGM for nite

dimensional spaces iteratively it turns out that any cohomology class comes from

a Fredholm representation asymptotically It suces for the Novikov conjecture

b ecause of nite dimensionality of manifolds

x Geometric interpretation

To indicate the geometric features of higher signatures let us consider signatures

of submanifoldsseeG Let b e a discrete group M b e a closed manifold and

f M K be a smo oth map Let us assume that K is realized by

a closed manifold V dimM dimV Then for a regular value m V the

cob ordism class of W f m is dened uniquely up to homotopy class of f

dimV

Moreover the Poincare dual class of W H M isf V where V H V

is the fundamental cohomology class Notice that the normal bundle of W is trivial

Thus

W LW W LM W LM f V M

W is a higher signature of M whichwenow dene as follows

Denition Let M b e a closed manifold and b e a discrete group Then a

higher signature of M is a characteristic number

LM f x M

where f M K is a continuous map and x H Q

It is conjectured that these characteristic numb ers are all homotopyinvariants

Let us see another geometric interpretation Let F X M b e a smo oth b er

bundle over M and assume F is k dimensional Then the at bundle induced from

the bration H M has a natural involution Thus H splits as H H H

and by the index theorem for families it follows

X LM chH H M

As a corollary we see that the right hand side is a homotopy invariant of b er

bundles over M seeAt

It is not necessary to construct a b er bundle corresp onding to each higher

signature To induce the homotopy invariance we only need a at bundle and

an involution over M From the point of view Lusztig succeeded in verifying

Novikov conjecture for free ab elian groups by the analytic metho d L Let Y

b e a compact top ological space and X be k dimensional compact manifold Let

Y X U p q b e a family of U p q representations of the fundamental

group of X Then one can construct a vector bundle E over Y X whichisatin

THE ASYMPTOTIC METHOD IN THE NOVIKOV CONJECTURE

the X direction E is admitted a non degenerate hermitian form and E splits

as E E E Using the splitting we obtain a family of quadratic forms

k k

H X E H X E C

Naturally there corresp onds X K Y which is homotopy invariant of X

Lusztig discovered the index theorem as follows Let Y X Y be the

pro jection Then

LX ch E E ch X

In particular we can takeas Y the representation space of the fundamental group of

X In the sp ecial case of the free ab elian group Z U the representation space

is the dual torus which is top ologically isomorphic to the torus T In the case of

a single U representation one can only obtain the signature However Lusztig

n n n

found the following There exist bases fa g and fb g of H T Zand H T Z

i i

suchthat

b ch X L LX f a X

i i

where f X T induces an isomorphism of the fundamental groups This is

enough to verify the Novikov conjecture for free ab elian groups In the case of

general noncommutative discrete groups the representation space will b e to o com

plicated and it will b e very dicult to apply this metho d to general noncommutative

discrete groups

x Fredholm representation

Mishchenko discovered the innite dimensional version of the metho d of at

vector bundles

Denition Let b e a discrete group Then a Fredholm representation of

is a set H H F where

H H are Hilb ert spaces

F H H isaF redholm map

U H H is a unitary representation suchthat F F is

i i i

a compact op erator for any

Using a Fredholm representation we can construct a virtual bundle over K

as follows From the condition we can construct an equivariantcontinuous map

f E B H H which satises

for some p oint x E f xF

for any p oints x y E f x f y is a compact op erator

Notice that f is unique up to homotopy Then the virtual bundle is f E

H E H and wewrite

f Fredholm representations g homotopy Virtual bundles over B

TKATO

Theorem Mishchenko Let f M B bea continuous map Then

LM f chF M

is an oriented homotopy invariant of M

Let us interpret this theorem as an innite version of the one of Lusztig By

doing surgery on the homology groups with lo cal co ecient wehave the resulting

homology only on the middle dimension Poincare duality on the homology gives a

symmetric form This is an elementoftheWall Lgroup L of the fundamen

tal group represented by a group ring valued nondegenerate symmetric matrix

If there is a unitary representation of then the matrix can be regarded as an

in vertible self adjoint op erator on an innite dimensional Hilb ert space The Fred

holm op erator F ofaFredholm representation decomp oses into an op erator valued

by matrix fF g corresp onding to the decomp osition of the Hilb ert space

ij ij

into p ositive and negative parts of the self adjoint op erators It turns out that the

diagonal parts F and F are also Fredholm op erators and F F are compact

op erators This follows essentially from the almost commutativityofthe unitary

representations and the Fredholm op erator in the denition of Fredholm represen

tation Thus we obtain a numb er indexF indexF Mishchenko discovered the

generalized signature theorem which asserts the coincidence of this numb er and the

characteristic numb er of the ab ove theorem The pro cess is parallel to the signature

theorem in the case of the simply connected spaces

x Novikov conjecture for word hyperbolic groups

It is natural to ask how large ch Fredholm representations is in H Q

By a celebrated work by AConnes MGromov and HMoscovici it is shown that

if is hyp erb olic then they o ccupyin H Q

In some cases of discrete groups Eilenb ergMaclane spaces are realized by com

pact smo oth manifolds In particular compact negatively curved manifolds them

selves are Eilenb ergMaclane spaces Hyp erb olic groups are intro duced by Gromov

The class is characterized by the essential prop erties which are p ossessed by the

fundamental groups of compact negatively curved manifolds Though the class is

very large they have reasonable classifying spaces which are enough to work instead

of Eilenb ergMaclane spaces at least for the Novikov conjecture The spaces are

called Rips complexes

Fact Let b e a discrete group Then there exists a family of nite dimensional

simplicial complexes fP g They satisfy the following

n n

acts on each P prop er discontin uously with compact quotient

if is torsion free then the action is also free

P P P

n n

ifishyp erb olic then P is contractible for suciently large n

In particular torsion free hyp erb olic groups have B represented by nite di

mensional simplicial complexes In the following we shall write P as a tubular

neighborhood in an emb edding P R P is an op en manifold with

n n

the induced metric from R In the following is a hyp erb olic group

THE ASYMPTOTIC METHOD IN THE NOVIKOV CONJECTURE

Kasparov KKgroups

Before explaining the metho d of CGM we shall quickly review Kasparovs KK

theory The KKgroups are used eectively to proveNovikov conjecture KK is a

bifunctor from a pair of distinct spaces X Y to ab elian groups whichiscovariant

on X and contravarianton Y The KKgroups include both K cohomology and

K homology

Roughly sp eaking K homology consists of the set of Dirac op erators on spaces

Precisely an elementofK X is represented byM E where

M is an even dimensional spin manifold which need not be compact or

connected

E is a complex vector bundle over M

is a prop er map from M to X

K X is the set of the ab ove triples quotiented by a certain equivalence relation

It is dual to K cohomology and the pairing is to take the index on twisted vector

bundles Let S b e the spin vector bundle over M and D S E S E

b e the Dirac op erator on M Then the pairing of K theory is FM E

indexD

E F

Fact There exists a Chern character isomorphism

inf

ch K X Q H X Q

inf

by ch E tdM M where H is the homology with lo cally nite innite

supp ort

Roughly sp eaking KKX Y is the set of sections over a family of elements of

K X over Y Thus if Y is a p oint

KK X pt K X

There is an analytical interpretation of top ological K homology Let C X bethe

set of the continuous functions on X vanishing at innity C X is C algebra

whose C norm is to take p ointwise supremum The analytical K homology K X

quotiented by an equivalence relation where is the set H H T

H is a Hilb ert space

C X B H isahomomorphism

i i

T H H is a b ounded op erator suchthat I T T I TT aT

T a are all compact op erators

The explicit map K X K X is to take L sections of twisted spin vector

bundles L M S E D Though D is an unb ounded op erator by making

E E

pseudo dierential calculus we can construct a b ounded op erator If M is compact

then it is D I D D As isapropermapitpullsback C X toC M

E E

and the homomorphism is the multiplication by a a C X If X is

a point then an element of K X is represented by a Fredholm op erator over

Hilb ert spaces K pt is naturally isomorphic to Z bytakingFredholm indices

KKpt Y is a family of Fredholm op erators over Y Thus

KK pt Y K Y

TKATO

Now let us dene the KKgroups First let us recall the denition of the analytical

K homology and consider the family version The set of sections over

the family of Hilb ert spaces over Y admits a natural C Y mo dule structure

As the homomorphism action is b erwise it commutes with that of C Y

A family of compact op erators will be formulated as an element of a norm

closure of nite rank pro jections in the set of endomorphisms of the C Y mo dule

So on we dene this precisely

Let us consider the triple E F where

E is a Z graded right C Y mo dule with a C Y valued inner pro duct It

is complete with resp ect to C norm of C Y E is called a Hilb ert mo dule ov er

C Y

is a degree homomorphism from C X to B E where B E isthe

set of C Y mo dule endomorphisms C X acts on E from the left

F B E is of degree suchthatF F a aFandF a

are all compact endomorphisms A compact endomorphism is an elementof B E

which lies in the closure of linear span of the rank one pro jections B E

z xyz We denote the set of compact endomorphisms by K E

If C Y itself is considered as C Y mo dule then B C Y is the set of

b ounded continuous functions on Y K C Y is also C Y

Let us denote the set of the ab ove triples E F by E X Y Notice that we

can replace C X and C Y byany C algebras A B and write E A B for the

set of triples which satisfy the ab ove replacing C X by A and C Y

by B

Now let us intro duce a homotopy equivalence relation as follows E F

is equivalenttoE F if there exists E F E A C B such that

E B f f F is isomorphic to E F where f C B B is

f i i i i i i

the evaluation maps

Denition KKX Y E X Y homotopy

It turns out that KKX Y is a group KKA B is dened similarly

n n

Notice that KKpt R is isomorphic to the K homology of R with compact

supp ort which is isomorphic to Z The generator of KKpt R is expressed using

n k

Cliord algebra Let n k be even Then byidentifying R with C anyvector

n k

in R acts on C by Cliord multiplication Then the generator is

n k

fC R C Fx g

jxj

in KKpt R

There is also equivariant KKtheory Let A and B admit automorphisms of

If X and Y are spaces then C X andC Y have natural actions Let

E A B b e the set of triples E F E A B such that there exists an action

of on E which satisfy

g a bgag gb gg g

agF g F is a compact endomorphism of E

Homotopy equivalence is dened similarly

THE ASYMPTOTIC METHOD IN THE NOVIKOV CONJECTURE

Denition KK X Y E X Y homotopy

Notice that KK pt pt is the set of Fredholm representations quotiented by

homotopy equivalence

There is a very imp ortant op eration in KKtheory called the intersection pro d

uct pairing see Bl

KK X Y KK Y Z KK X Z

Lipschitz geometry

In the essential step CGM constructs an element KK pt P Roughly

sp eaking the construction is as follows

First of all using the contractibilityof P one constructs a map which induces

Poincare duality

P P T P

n n n

inf

Namely for H P Q z H P Q

n n

inf

H P Q H P Q

n n

Prop osition CGM If is berwise proper Lipschitz then one can construct

Let P P T P P R b e the b erwise prop er Lipschitz map

n n n n

which induces Poincare duality Let us take e P and restrict on P e Then

n n

fC P C g

jxj

in KK pt P is the desired one If is not b erwise Lipschitz then the ab ove

do es not dene an element of the equivariant KKgroup To ensure F F is

a compact endomorphism it is enough to see that jF x F xj go es to zero

when x go es to innity This follows by simple calculation from the Lipschitzness

A priori we only have a b erwise prop er map which induces Poincare duality

It is natural to try to deform the map so that it b ecomes b erwise prop er Lipschitz

by a prop er homotopy To do so rst using the hyp erb olicitywehave the following

map

Prop osition Let us take a suciently lar ge n and a suciently smal l

constant The there exists a map

F P P P P

n n n n

such that

F is berwise Lipschitz

TKATO

there exists a berwise proper homotopy F which connects F to the identity

Remark The existence of such F implies that P mustbecontractible

Let us take suciently large r and put D fx y P P dx y r g By

n n

mo difying slightlywemay assume that F j j

F D F D

Let us put

B fx y P P F x y F F Fx y D g

i n n

D B B

i i i

Let us dene

P P T P

n n n

i i

j F

It is not dicult to see that is b erwise prop er Lipschitz and it is b erwise

prop er homotopic to

Using the Kasparovintersection pro duct wehave a map

KK P pt KK pt pt

x x

Theorem CGM There exists the fol lowing commutative diagram

KK P Q K B

y y

inf

H P H B

where PD is the Poincare duality

In the case of cohomology groups of o dd degrees we can reduce to the case of

even ones by considering Z Thus

Corollary Let beahyperbolic group and f M K bea continuous

map Then LM f x M is an oriented homotopy invariant for any x

H Q Namely let p M M be an oriented homotopy equivalence Then

LM p f x M LM f x M

Notice that in the case of hyp erb olic groups wehave used the fact that K

w as realized by a nite dimensional simplicial complex over Q On the other hand

we cannot exp ect it on more large classes of discrete groups in particular quasi

geo desic bicombing groups whichwe shall treat in the next section For the class

we cannot exp ect even that the ranks of cohomology over Q are nite

THE ASYMPTOTIC METHOD IN THE NOVIKOV CONJECTURE

x Quasi geodesic bicombing groups

In ECHLPT a very large class of discrete groups is dened The elements

of the class are called combing groups It contains hyp erb olic groups and quasi

geo desic bicombing groups dened later

Theorem ECHLPT If is a combable group then K space can be

realizedbyaCW complex such that the number of cel ls in each dimension is nite

As an immediate corollary of this we can see that dimH Q for each

n Using this fact in the following construction we shall make an analogy of the

case of hyp erb olic groups on spaces whichapproximates K

A set of generators of a discrete group determines a dimensional simplicial

complex called Cayley graph G G has a natural metric Notice that the

universal covering spaces of non p ositively curved manifolds have the convex prop

erty With this in mind we shall dene the following

Denition G If has the following prop erties we call it a bicombing

group Let us x a generating set of Then there exists a continuous and

equivariantmap

S G

such that for some k C it satises

S S

dS S k td td C

t t

Though S G connects and we shall require bal

anced curves

Denition G Let b e bicombing Wesa y that is b ounded if for some

k C it satises

k jt t jd C dS S

t t

Denition b ounded bicombing is quasi geo desic if for every a su

ciently small and tst S e t S e s Moreover let us

denote a unit sp eed path of S e by

jS e j G Then for d

Ud sup d t t C

jS e jk j j C

Furthermore for some AB S satises

d S Atd B

Using S it is easy to prove the following lemma

TKATO

Lemma Al If is a quasi geodesic bicombing group then each Rips complex

P is contractible in P for large n ni

i n

From this we see that for a torsion free quasi geo desic bicombing group

P lim P is a realization of E Unlike to the case of hyp erb olic groups

i i

we cannot make F P P P P as b efore b ecause P is not

n n n n n

contractible in itself However w ehave the following family of maps

Prop osition Let us take an arbitrary family of smal l constants f g

i i

Then for some family of Rips complexes fP g

i i i

there exists a family of maps

F P P P P

n ni n ni

such that

F is berwise Lipschitz

i i

F is berwise proper homotopic to the inclusion

Let P P R b e a b erwise prop er map which induces Poincare

n n

duality Using the ab ove family of maps we can construct the following commuta

tive diagram of maps

R P P

incl incl

y y

P R P

incl incl

y y

To pro duce a prop er Lipschitz map we need to control growth of these maps

at innity In this case we can construct which satisfy the following There

exist families of constants fC g fa g such that the Lipschitz constant of on

i i i

N r fx y jx P y P d x y r g for suciently large r is b ounded by

i i i

C H a r whereH is a Lipschitz function on

i i

Prop osition Using these maps we have

P R P

n n

which is berwise proper homotopic to Moreover let

pr R R N dimP

Then pr is berwise proper Lipschitz

Let us recall that homology commutes with spaces under the direct limit op era

tion Thus H P lim H P With the fact that the rank of H P

n n N n

is nite for every N wehave

THE ASYMPTOTIC METHOD IN THE NOVIKOV CONJECTURE

Lemma Let us take any large N Then for Nthere exists n such that

i H P Q H P Q

is surjective where i P P is the inclusion

As b efore we can construct KK pt P

Theorem The fol lowing diagram commutes

KK P pt K B

i ch

y y

inf

H P H P

n n

We cannot construct the homotopybetween pr and through the map to

R Let K N and Z P P Toshow the commutativity of the diagram

n n

we construct a homotopybetween the following two elements in KKpt Z

fC Z C F x g

j xj

pr x

pr fC Z C F x g

jpr xj

Let C Then there are natural inclusions which

N N N

preserves the metrics Let C b e the innite dimensional Hilb ert space which

is the completion of the union By adding degenerate elements we can express

fC Z C F G g

pr fC Z C F G g

through maps to R wecan Using the prop er homotopybetween and pr

construct the homotopybetween the elements in KKpt Z

Corollary Let be a torsion free quasi geodesic bicombing group Then for

arbitrary large N and x H B Q there exists a Fredholm representation F

such that x chF H B Q N

Corollary For torsion freequasigeodesic bicombing groups the higher sig

natures are oriented homotopy invariants

References

Al J Alonso Combings of groups in Algorithms and Classication in Combinatorial

Group theoryvol MSRI Publ pp

At MAtiyah The signature of ber bund les Global analysis in honor of KKo daira

Princeton University Press Princeton

Bl B Blackadar K theory for operator algebras MSRI Publ vol

TKATO

CGM A Connes M Gromov H Moscovici Group cohomology with Lipschitz Control l and

Higher Signatures Geometric and Functional Analysis

ECHLPT Epstein DBA Cannon JW Holt DF LevySVF Paterson MS and Thurston

WP Wordprocessing and group theory Jones and Bartlett Publ

G M Gromov Asymptotic invariants of innite groups London Math So c Lecture

Note Series

G MGromov Positive curvature macroscopic dimension spectral gaps and higher sig

naturesFunctional Analysis on the Eve of the st Century Pro c Conf in honor

of IMGelfands th birthday SGindikin JLep owsky and RWilson eds

K TKato Asymptotic Lipschitz cohomology and higher signatures Geometric and Func

tional Analysis

L GLusztig Novikovs higher signature and families of el liptic operators JDierential

Geometry

Mis AS Mishchenko Innite dimensional representations of discrete groups and higher

signature Izv Ak ad Nauk SSSR Ser Mat

Department of Mathematics Faculty of Science Kyoto University Kyoto

Japan

Email address tkatokusmkyotouacjp

A NOTE ON EXPONENTIALLY NASH G

MANIFOLDS AND VECTOR BUNDLES

TomohiroKawakami

Department of Lib eral Arts Osaka Prefectural

College of Technology NeyagawaOsakaJapan

Intro duction

Nash manifolds have b een studied for a long time and there are many brilliant

works eg

n n

The semialgebraic subsets of R are just the subsets of R denable in the

standard structure R R of the eld R of real numb ers

stan

However any nonp olynomially b ounded function is not denable in R where

stan

an a p olynomially b ounded function means a function f R R admitting

integer N N and a real number x R with jf xj x x x C Miller

proved that if there exists a nonp olynomially b ounded function denable in

an ominimal expansion R ofR then the exp onential function

stan

exp R R is denable in this structure Hence R R exp is

exp

a natural expansion of R There are a number of results on R eg

stan exp

Note that there are other structures with prop erties similar

to those of R

exp

r r

Wesay that a C manifold r isan exponential l y C N ash manif ol d

if it is denable in R See Denition Equivariantsuch manifolds are dened

exp

in a similar way See Denition

In this note we are concerned with exp onentially C Nash manifolds and equi

variant exp onentially C Nash manifolds

Theorem Any compact exponential ly C Nash manifold r admits

an exponential ly C Nash imbedding into some Euclidean space

Note that there exists an exp onentially C Nash manifold which do es not admit

any exp onentially C imb edding into any Euclidean space Hence an exp onen

tially C Nash manifold is called affine if it admits an exp onentially C Nash

imb edding into some Euclidean space See Denition In the usual Nash cat

egory Theorem is a fundamental theorem and it holds true without assuming

compactness of the Nash manifold

Equivariant exp onentially Nash vector bundles are dened as well as Nash ones

See Denition

Mathematics Subject Classication P P P S S A

TOMOHIROKAWAKAMI

Theorem Let G beacompact ane Nash group and let X beacompact ane

exponential ly C Nash G manifold with dim X Then for any C G vector

bund le of positive rank over X there exist two exponential ly C Nash G vector

bund le structures of such that they are exponential ly C Nash G vector bund le

isomorphic but not exponential ly C Nash G vector bund le isomorphic

be Theorem Let G bea compact ane exponential ly Nash group and let X

a compact C G manifold If dim X and dim X then X admits two

exponential ly C Nash G manifold structures which are exponential ly C Nash G

dieomorphic but not exponential ly C Nash G dieomorphic

In the usual equivariant Nash categoryany C Nash G vector bundle isomor

phism is a C Nash G one and moreover every C Nash G dieomorphism is a

C Nash G one Note that Nash structures of C G manifolds and C G vector

bundles are studied in and resp ectively

In this note all exp onentially Nash G manifolds and exp onen tially Nash G vector

bundles are of class C and manifolds are closed unless otherwise stated

Exp onentially Nash G manifolds and exp onentially Nash G vector

bundles

Recall the denition of exp onentially Nash G manifolds and exp onentially Nash

G vector bundles and basic prop erties of exp onentially denable sets and exp o

nentially Nash manifolds

Denition An R ter m is a nite string of symb ols obtained by rep eated

exp

applications of the following two rules

Constants and variables are R terms

exp

If f is an mplace function symbol of R and t t are R terms then

exp m exp

the concatenated string f t t isanR term

m exp

An R f or mul a is a nite string of R terms satisfying the following three

exp exp

rules

For anytwo R terms t and t t t and t t are R formulas

exp exp

If and are R formulas then the negation the disjunction and

exp

the conjunction are R formulas

exp

If is an R formula and v is a variable then v and v are R

exp exp

formulas

An exponential l y def inabl e set X R is the set dened byanR formula

exp

with parameters

n m

A map f X Y Let X R and Y R b e exp onentially denable sets

n m

is called exponential l y def inabl e if the graph of f R R is exp onentially

denable

On the other hand using any exp onentially denable subset of R is the

n m n

image of an R semianalytic set by the natural pro jection R R R for

some m Here a subset X of R is called R semianal y tic if X is a nite union of

sets of the following form

fx R jf xg x i k j l g

i j

where f g Rx x expx expx

i j n n

The following is a collections of prop erties of exp onentially denable sets cf

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

Prop osition cf Any exponential ly denable set consists of only

nitely many connectedcomponents

n m

Let X R and Y R be exponential ly denable sets

The closure ClX and the interior IntX of X areexponential ly denable

The distance function dx X from x to X denedbydx X inf fjjx y jjjy

X g is a continuous exponential ly denable function where jjjj denotes the standard

norm of R

Let f X Y be an exponential ly denable map If a subset A of X is

exponential ly denable then so is f A and if B Y is exponential ly denable

then so is f B

Let Z R beanexponential ly denable set and let f X Y and h Y

Z be exponential ly denable maps Then the composition h f X Z is also

exponential ly denable In particular for any two polynomial functions f g R

g xf x

R ff g R denedbyhxe is exponential ly R the function h

denable

The set of exponential ly denable functions on X forms a ring

Any two disjoint closed exponential ly denable sets X and Y R can be

separatedbyacontinuous exponential ly denable function

n m r

Let U R and V R b e op en exp onentially denable sets A C r

map f U V is called an exponential l y C N ash map if it is exp onentially

r r

denable An exp onentially C Nash map g U V is called an exponential l y C

N ash dif f eomor phism if there exists an exp onentially C Nash map h V U

such that g h id and h g id Note that the graph of an exp onentially C

Nash map may b e dened byanR formula with quantiers

exp

Theorem Let S S R be exponential ly denable sets Then

d n

there exists a nite family W f g of subsets of R satisfying the fol lowing four

conditions

n d d d d

and S f S g for i k are disjoint R j

i i d

Each is an analytic cel l of dimension d

d e

is a union of some cel ls with ed

d d e d e e

then satises Whitneys conditions a If W

and b at al l points of

Theorem allows us to dene the dimension of an exp onentially denable set

E by

dim E maxfdim j is an analytic submanifold contained in E g

Example The C function R R dened by

if x

e if x

is exp onentially denable but not exp onentially Nash This example shows that

an exp onentially denable C map is not always analytic This phenomenon do es

not o ccur in the usual Nash category We will use this function in section

TOMOHIROKAWAKAMI

The Zariski closure of the graph of the exp onential function exp R R in

R is the whole space R Hence the dimension of the graph of exp is smaller than

that of its Zariski closure

The continuous function h R R dened by

e if n x n

hx for n Z

e if n x n

is not exp onentially denable but the restriction of h on any b ounded exp onentially

denable set is exp onentially denable

Denition Let r b e a nonnegativeinteger or

r r

An exponential l y C N ash manif ol d X of dimension d is a C manifold

admitting a nite system of charts f U R g such that for each i and j U

i i i i

U is an op en exp onentially denable subset of R and the map j U U

j j i i j

U U U U is an exp onentially C Nash dieomorphism Wecall

i i j j i j

these charts exponential l y C N ash A subset M of X is called exponential l y

def inabl e if every U M is exp onentially denable

i i

n r

An exp onentially denable subset of R is called an exponential l y C Nash

r n

submanif ol d of dimension d if it is a C submanifold of dimension d of R An

r r

exp onentially C r Nash submanifold is of course an exp onentially C Nash

manifold

r r

Let X resp Y b e an exp onentially C Nash manifold with exp onentially C

n m r

Nash charts f U R g resp f V R g A C map f X Y

i i i j j j

is said to b e an exponential l y C Nash map if for any i and j f V U is

i j i

op en and exp onentially denable in R and that the map f f V

j i j

m r

U R is an exp onentially C Nash map

Let X and Y b e exp onentially C Nash manifolds Wesaythat X is exponen

r r

tial l y C N ash dif f eomor phic to Y if one can nd exp onentially C Nash maps

f X Y and h Y X suchthat f h id and h f id

An exp onentially C Nash manifold is said to b e affine if it is exp onentially

r r l

C Nash dieomorphic to some exp onentially C Nash submanifold of R

A group G is called an exponential l y N ash g r oup resp an af f ine exponen

tial l y N ash g r oupifG is an exp onentially Nash manifold resp an ane exp o

nentially Nash manifold and that the multiplication G G G and the inversion

G G are exp onentially Nash maps

Denition Let G b e an exp onentially Nash group and let r

An exp onentially C Nash submanifold in a representation of G is called an

exponential l y C N ash G submanif ol d if it is G invariant

r r

An exp onentially C Nash manifold X is said to b e an exponential l y C Nash

G manif ol d if X admits a G action whose action map G X X is exp onentially

C Nash

r r

Let X and Y b e exp onentially C Nash G manifolds An exp onentially C Nash

map f X Y is called an exponential l y C Nash G map if it is a G map An

r r

exp onentially C Nash G map g X Y is said to b e an exponential l y C Nash

G dif f eomor phism if there exists an exp onentially C Nash G map h Y X

suchthat g h id and h g id

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

Wesay that an exp onentially C Nash G manifold is affine if it is exp onentially

r r

C Nash G dieomorphic to an exp onentially C Nash G submanifold of some

representation of G

Wehave the following implications on groups

an algebraic group an ane Nash group an ane exp onentially Nash group

an exp onentially Nash group a Lie group

Let G b e an algebraic group Then we obtain the following implications on G

manifolds

a nonsingular algebraic G set an ane Nash G manifold

an ane exp onentially Nash G manifold an exp onentially

Nash G manifold a C G manifold

ys an ane exp onentially Nash Moreover notice that a Nash G manifold is not alwa

G manifold

In the equivariant exp onentially Nash categorythe equivariant tubular neigh

b orho o d result holds true

Prop osition Let G bea compact ane exponential ly Nash group and let

X be an ane exponential ly Nash G submanifold possibly with boundary in a repre

sentation of G Then there exists an exponential ly Nash G tubular neighborhood

U p of X in namely U is an ane exponential ly Nash G submanifold in

and the orthogonal projection p U X is an exponential ly Nash G map

Denition Let G b e an exp onentially Nash group and let r

r r

A C G vector bundle E p X of rank k is said to be an exponential l y C

N ash G v ector bundl e if the following three conditions are satised

a The total space E and the base space X are exp onentially C Nash G

manifolds

b The pro jection p is an exp onentially C Nash G map

c There exists a family of nitely many lo cal trivializations fU U

i i i

R p U g such that fU g is an op en exp onentially denable

i i i i

covering of X and that for any i and j the map jU U R

j i j

k k r

U U R U U R is an exp onentially C Nash map

i j i j

We call these lo cal trivializations exponential l y C N ash

Let E p X resp F q X b e an exp onentially C Nash G vector

bundle of rank n resp m Let fU U R p U g resp fV

i i i i i j j

r m

tially C Nash lo cal trivializations of resp V R q V g b e exp onen

j j j

r r

A C G vector bundle map f is said to b e an exponential l y C Nash

G vector bundl e map if for any i and j the map f jU V R

j i i j

n m r r

U V R U V R is an exp onentially C Nash map A C G

i j i j

r n

section s of is called exponential l y C N ash if each sjU U U R

i i i

is exp onentially C Nash

Two exp onentially C Nash G vector bundles and are said to b e exponen

TOMOHIROKAWAKAMI

r r

tial l y C N ash G v ector bundl e isomor phic if there exist exp onentially C Nash G

vector bundle maps f and h suchthat f h id and h f id

Recall universal G vector bundles cf

Denition Let b e an ndimensional representation of G and B the represen

tation map G GL R of Supp ose that M denotes the vector space of

n nmatrices with the action g A G M B g AB g M For

an y p ositiveinteger k we dene the vector bundle k E kuGk

as follows

GkfA M jA A A A TrA k g

E kfA v Gk jAv v g

u E k GkuA v A

where A denotes the transp osed matrix of A and TrA stands for the trace of A

Then k is an algebraic set Since the action on k is algebraic it is an

algebraic G vector bundle We call it the univ er sal G v ector bundl e associated

with and k Since Gk and E k are nonsingular k is a Nash G

vector bundle hence it is an exp onentially Nash one

Denition An exp onentially C Nash G vector bundle E p X ofrank

k is said to b e str ong l y exponential l y C N ash if the base space X is ane and

that there exist some representation of G and an exp onentially C Nash G map

f X Gksuchthat is exp onentially C Nash G vector bundle isomorphic

to f k

Let G b e a Nash group Then wehave the following implications on G vector

bundles over an ane Nash G manifold

aNashG vector bundle an exp onentially Nash G vector bundle a

C G vector bundle and

a strongly Nash G vector bundle a strongly exp onentially Nash G vector

bundle an exp onentially Nash G vector bundle

Pro of of results

n n

A subset of R is called locally closed if it is the intersection of an op en set R

and a closed set R

To prove Theorem we recall the following

Prop osition Let X R be a local ly closed exponential ly denable set

and let f and g becontinuous exponential ly denable functions on X with f

g Then there exist an integer N and a continuous exponential ly denable

function h X R such that g hf on X In particular for any compact

subset K of X there exists a positive constant c such that jg jcjf j on K

Proof of Theorem Let X b e an exp onentially C Nash manifold If dim X

then X consists of nitely manypoints Thus the result holds true

m l r

dim X Let f U R g be exp onentially C Nash Assume that

i i

charts of X Since X is compact shrinking U if necessarilywemay assume that i

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

every U is the op en unit ball of R whose center is the origin Let f be

i i

U U the function on R dened by f xjjxjj Then f

i i i i

Hence replacing the graph of f on U by U each U is closed in R

i i i i i i

m m m

Consider the stereographic pro jection s R S R R Comp osing

and swehave an exp onentially C Nash imb edding U R such that

i i

the image is b ounded in R and

U U

i i i i

consists of one p oint say Set

m m

X X

k k m m

x x x x R R x x

m m

j j

g U R

i i i

b edding of for a suciently large integer k Then g is an exp onentially C Nash im

U into R Moreover the extension g X R of g dened by g onX U

i i i i i

Wenowprove that g is of class exp onentially C Nash It is sucient to see this on

eachexponentially C Nash co ordinate neighborhood of X Hence wemay assume

that X is op en in R We only havetoprove that for any sequence fa g in U

j i

convergenttoa pointofX U and for any N with j j r fD g a g

i i j

P P

m m

k k

converges to On the other hand g where

i i im

ij ij

j j

Each is b ounded and every f a g converges to

i i im ij ij i

i i

zero and

k k

j j D D j jD

is is

ij ij

k l

j C D D D jC j j

ij ij is

ij ij

where C C are constants and is the p ositive continuous exp onentially denable

function dened by

maxf jD x D xD xjg x

ij ij is

Dene

minfj xj xg on U on U

ij i ij i

ij ij

on X U on X U

i i

Then and are continuous exp onentially denable functions on X suchthat

ij ij

X U

i ij

Hence by Prop osition wehave j jd on some op en exp onentially den

ij ij

able neighborhood V of X U in X for some integer l where d is a constant i

TOMOHIROKAWAKAMI

On the other hand by the denition of j j Hence the ab ove argument

ij ij

proves that

k r l

jD jc j j

on U V where c is a constant and we take k such that k r l Hence

each g is of class exp onentially C Nash It is easy to see that

g X R

is an exp onentially C Nash imb edding

By the similar metho d of wehave the following

Theorem Let G beacompact ane exponential ly Nash group and let X

beacompact ane exponential ly Nash G manifold

For every C G vector bund le over X there exists a strongly exponential ly

Nash G vector bund le which is C G vector bund le isomorphic to

For any two strongly exponential ly Nash G vector bund les over X they are

exponential ly Nash G vector bund le isomorphic if and only if they are C G vector

bund le isomorphic

We prepare the following results to prove Theorem

Prop osition L et M be an ane exponential ly Nash G manifold in a

representation of G

The normal bund le L q M in realizedby

L fx y M jy is orthogonal to T M gq L M qx y x

is an exponential ly Nash G vector bund le

If M is compact then some exponential ly Nash G tubular neighborhood U of M

in obtainedbyProposition is exponential ly Nash G dieomorphic to L

Prop osition Let G bea compact ane exponential ly Nash group and let

E p Y be an exponential ly Nash G vector bund le of rank k over an ane

exponential ly Nash G manifold Y Then is strongly exponential ly Nash if and

only if E is ane

Lemma Let D and D be open bal ls of R which have the same center x

and let a resp bbe the radius of D resp D with ab Suppose that A and

B are two real numbers Then there exists a C exponential ly denable function

n n

f on R such that f A on D and f B on R D

Proof We can assume that A B and x

At rst we construct such a function when n Then we mayassumethat

D a aandD b bbeopenintervals Recall the exp onentially denable

C function dened in Example The function R R dened by

xb xb xb xb xx a

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

is the desired function Therefore f R Rfx jxj is the required

function where jxj denotes the standard norm of R

proof of Theorem By Theorem wemay assume that is a strongly exp o

nentially C Nash G vector bundle We only have to nd an exp onentially C Nash

G vector bundle which is exp onentially C Nash G vector bundle isomorphic to

but not exp onentially C Nash G vector bundle isomorphic to

As well as the usual equivariant Nash category X is an exp onentially Nash G

submanifold of X Take an op en exp onentially denable subset U of X such that

jU is exp onentially C Nash vector bundle isomorphic to the trivial bundle and

G G

that X U Since dim X there exists a onedimensional exp onentially

Nash G submanifold S of U which is exp onentially Nash dieomorphic to the unit

circle S in R Moreover there exist twoopenG invariant exp onentially denable

subsets V and V of U such that V V S and V V consists of two op en balls Z

Nash G vector bundle E rV V and Z We dene the exp onentially C

over V V to b e the bundle obtained by the co ordinate transformation

I on Z

g V V GLg

I on Z

where I denotes the unit matrix is suciently small and stands for the

b er of jU This construction is inspired by the pro of of

Let V p V i b e exp onentially Nash G co ordinate functions

i i i

Consider an extension of the exp onentially C Nash section f on S V dened of

by f xx I If we extend f through Z then the analytic extension f

to S V satises f x I x S V However the analytic extension f to

f x I Thus the smallest analytic set S V through Z satises

containing the graph of f spins innitely over S Hence jS is not exp onentially

C Nash G vector bundle isomorphic to jS By Theorem jS is not strongly

exp onentially C Nash Thus the exp onentially C Nash G vector bundle over

X obtained by replacing jV V by is not exp onentially C Nash G vector

bundle isomorphic to

On the other hand by Lemma we can construct an exp onentially C Nash

G map H from a G invariant exp onentially denable neighb orho o d of U X in

I and H I outside of some G invariant U to GL such that H jZ

exp onentially denable neighb orho o d of Z Since is suciently small using this

map we get an exp onentially C Nash G vector bundle isomorphism

Proof of Theorem By the pro of of Theorem X is C G dieomorphic

to some ane exp onentially Nash G manifold Hence wemay assume that X is an

ane exp onentially C Nash G manifold

Since X is an exp onentially C Nash G submanifold of X there exists an

exp onentially Nash G tubular neighb orho o d T qof X in X b y Prop osition

Moreover wemay assume that T is exp onentially C Nash G dieomorphic to the

total space of the normal bundle of X in X b ecause of Prop osition Note

that is a strongly exp onentially C Nash G vector bundle over X and that each

b er is a representation of G Take an op en G invariant exp onentially denable

subset U of X such that jU is exp onentially C Nash G vector bundle isomorphic

to the trivial bundle U where denotes the b er of jU

TOMOHIROKAWAKAMI

By the pro of of Theorem there exists an exp onentially C Nash G vector

bundle over U suchthat is not exp onentially C Nash G vector bundle iso

morphic to jU and that there exists an exp onentially C Nash G vector bundle

isomorphism H jU suchthat H is the identityoutside of some op en G

invariantexponentially denable set

Replacing the total space of jU by that of we have an exp onentially C

Nash G manifold Y which is not exp onentially C Nash G dieomorphic to X

Moreover using H one can nd an exp onentially C Nash G dieomorphism from

X to Y

Note that Y is not exp onentially C Nash G ane but exp onentially C Nash

G ane by Prop osition

Remarks

It is known in that every compact Lie group admits one and exactly one

algebraic group structure up to algebraic group isomorphism Hence it admits an

ane Nash group structure Notice that all connected onedimensional Nash groups

and lo cally Nash groups are classied by and resp ectively In particular

the unit circle S in R admits a nonane Nash group structure

But the analogous result concerning nonane exp onentially Nash group struc

tures of centerless Lie groups do es not hold

Remark Let G be a compact centerless Lie group Then G does not admit

any nonane exponential ly Nash group structure

Proof Let G b e an exp onentially Nash group which is isomorphic to G as a Lie

group Then the adjoint representation Ad G Gl R is exp onentially den

able bythesimilar metho d of Lemma and it is C where n denotes the

dimension of G Hence Ad is an exp onentially Nash one and its kernel is the center

of G Therefore the image GofAd is an ane exp onentially Nash group and Ad

is an exp onentially Nash group isomorphism from G to G

It is known that anytwodisjoint closed semialgebraic sets X and Y in R can

R namely there exists a C Nash be separated by a C Nash function on

function f on R such that

fonX and fonY

The following is a weak equivariant version of Nash category and exp onentially

Nash category

Remark Let G be a compact ane Nash resp a compact ane exponen

tial ly Nash group Then any two disjoint closed G invariant semialgebraic resp

disjoint closed G invariant exponential ly denable sets in a representation of G

can be separated by a G invariant continuous semialgebraic resp a G invariant

continuous exponential ly denable function on

Proof By the distance dx X ofx between X is semialgebraic resp exp onentially

denable Since G is compact dx X is equivariant Hence F RFx

dx Y dx X is the desired one

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

Remark Under the assumption of if one of the above two sets is com

pact then they are separatedbyaG invariant entirerational function on where

an entirerational function means a fraction of polynomial functions with nowhere

vanishing denominator

Proof Assume that X is compact and Y is noncompact Let s S R

b e the stereographic pro jection and let S fg Since X is compact sX

and sY fg are compact and disjoint Applying Remark we have a G

invariant continuous semialgebraic resp a G invariant continuous exp onentially

denable function f on R By the classical p olynomial approximation theorem

and Lemma we get a G invariant p olynomial F on R such that F jS

is an approximation of f Since sX and sY fg are compact F s is the

required one

Remark Let X R beanopen resp a closed exponential ly denable set

Suppose that X is a nite union of sets of the fol lowing form

fx R jf x f xg x g x g

i j

resp fx R jf x f xg x g x g

i j

where f f and g g areexponential ly Nash functions on R Then X is

i j

a nite union of sets of the fol lowing form

fx R jh x h x g

resp fx R jh x h x g

where h h areexponential ly Nash functions on R

Note that any exp onentially denable set in R can b e describ ed as a nite union

of sets of the following form

fx R jF x F xG x G x g

s t

Here eachof F F and G G is an exp onentially Nash function dened on

s t

er its domain is not always some op en exp onentially denable subset of R howev

the whole space R

We dene exp x for n N and x R by exp x x and exp x

n n

expexp x The following is a b ound of the growth of continuous exp onentially

denable functions

Prop osition Let F be a closed exponential ly denable set in R and let

f F R be a continuous exponential ly denable function Then there exist

cnm N such that

xjc exp jjxjj for any x F jf

where jj jj denotes the standard norm of R

Proof of Remark It suces to provethe result when X is op en b ecause the

other case follows by taking complements

TOMOHIROKAWAKAMI

Let

B fx R jf x f xg x g x g

u v

where all f and all g are exp onentially Nash functions on R Set f f f

i j

By Prop osition and g x jg xj g x On R X g xiff x

i i

there exists an integer N and a continuous exp onentially denable function h

n N n

on R X such that g hf on R X By Prop osition wehave some c R

m n

and some m n N such that jhxjc exp jjxjj onR X Dene B

N n m m

g x g x fx R jcf x exp jjxjj g x g

i n m

Then B B X Therefore replacing B by B wehave the required union

References

C Chevalley Theory of Lie groups Princeton Univ Press Princeton N J

M Coste and M Shiota Nash triviality in families of Nash manifoldsInvent Math

M Coste and M Shiota Thoms rst isotopy lemma a semialgebraic version with uniform

bound Real analytic and algebraic geometry Pro ceedings of the International Conference held

at Trento Italy Walter de Gruyter Berlin New York

KH Dovermann M Masuda and T Petrie Fixedpoint free algebraic actions on varieties

dieomorphic to R Progress in Math Birkhauser

L van den Dries A Macintyre and D Marker The elementary theory of restricted analytic

elds with exponentiation Ann of Math

al elds and psedonite elds Israel J E Hrushovski and A Pillay Groups denable in loc

Math

T Kawakami Algebraic G vector bund les and Nash G vector bund les Chinese J Math

T Kawakami Exponential ly Nash manifolds and equivariant exponential ly Nash manifolds

preprint

T Kawakami Nash G manifold structures of compact or compactiable C G manifoldsJ

Math So c Japan

SKoike TFukui and MShiota Modied Nash triviality of family of zerosets of real poly

nomial mappings preprint

TL Loi Analytic cel l decomposition of sets denable in the structure R AnnPol Math

exp

TL Loi C regular points of sets denable in the structure R exp preprint

TL Loi On the global Lojasiewicz inequalities for the class of analytic logarithmicoexponen

tial functions CR Acad Sci Paris

TL Loi Whitney stratications of sets denable in the structure R toappear

exp

AN Nesin and A Pillay Some model theoryofcompact Lie groupsTrans Amer Math So c

JJ Madden and CH Stanton Onedimensional Nash groupsPacic J Math

C Miller Exponentiation is hardtoavoid Pro c Amer Math So c

T Mostowski Some properties of the ring of Nash functions Ann Scu Norm Sup Pisa

M Shiota Abstract Nash manifolds Pro c Amer Math So c

M Shiota Approximation theorems for Nash mappings and Nash manifoldsTrans Amer

Math So c

M Shiota Classication of Nash manifolds Ann Inst Fourier Grenoble

to app ear M Shiota Geometry of subanalytic and semialgebraic sets

M Shiota Nash manifolds Lecture Note in Mathematics vol SpringerVerlag

A Tarski Adecision method for elementary algebra and geometry nd ed revised Berkely and Los Angels

EXPONENTIALLYNASHG MANIFOLDS AND VECTOR BUNDLES

A Wilkie Model completeness results for expansions of the real eld restricted Pfaan

functions to app ear

A Wilkie Model completeness results for expansions of the real eld the exponential

function to app ear

ON FIXED POINT DATA OF SMOOTH ACTIONS ON SPHERES

Masaharu Morimoto

We rep ort results obtained jointly with K Pawalowski

There are two fundamental questions ab out smo oth actions on manifolds Let

G b e a nite group and M a manifold

Question Which manifolds F can b e the Gxed p oint sets of Gactions on M

ie M F

Question Which Gvector bundles over F can b e the Gtubular neighb orho o ds

ie Gnormal bundles of F M in M

If can b e realized as a subset of M in the wayabove wesaythatF o ccurs as

G W

the Gxed p ointdatain M If for a real Gmo dule W with W F

o ccurs as the Gxed p oint data in M then wesaythatF stably o ccurs as the

Gxed point data in M These questions were studied by B Oliver O in the

case where G is not of prime power order and M is a disk or a Euclidean space

The topic of the currenttalk is the case where G is an Oliver group and M is a

sphere

Let G be a nite group not of prime power order A Gaction on M is called

P G

P prop er if M M for any Sylow subgroup P of G There are necessary

conditions for F to stably o ccur as the Gxed point data of a P prop er G

action on a sphere

F Oliver Condition F M mo d n where n is the integer called

G G

Olivers numb er O

B Pro duct Bundle Condition in KOF

B Smith Condition For each prime p and any Sylow psubgroup P of G

in KO F

F P

By Oliver O Conditions F B and B are also necessarysucient condi

tions for F to stably o ccur as the Gxed p oin t data in a disk By O n is

equal to if and only if there are no normal series PHG suchthat jP j p HP

is cyclic and jGH j q s t A group G with n is called an Oliver

group Clearly any nonsolvable group is an Oliver group A nilp otent group is an

Oliver group if and only if it has at least three noncyclic Sylow subgroups In the

case where G is an Oliver group Condition F provides no restriction

We b egin the preparation for our sucient conditions For a nite group G and

p p

aprimepletG denote the minimal normal subgroup of G suchthat GG is of

MASAHARU MORIMOTO

pp ower order p ossibly G G Let LG denote the set of all subgroups H of G

such that H G for some prime p Let P G denote the set of all subgroups P

of G suchthat jP j is a prime p ower p ossibly jP j A Gaction on M is said to

be P Lprop er if the action is P prop er and if any connected comp onentof X

H LG do es not contain a connected comp onentof M as a prop er subset If

G is an Oliver group then the Gaction on

V GRG R RGG R

pnjGj

is P Lprop er LM A nite group G is said to b e admissible if there is a real

P H

Gmo dule V such that dim V dimV for any P PGandany H G with

H P and dim V for any H LG

Theorem MMM Yanagihara MY Let G be an Oliver group If G G

or G G for at least distinct odd primes then G is admissible In particular

an Oliver group G is admissible in each case G is nilpotent G is perfect

The symmetric group of degree is not admissible

K H DovermannM Herzog recently proved that S n are admissible

Our main result is

Theorem A Let G bean admissible Oliver group resp an Oliver group Let

F be a closed manifold resp a nite discrete space and let be a real Gvector

bund le over F such that dim whenever H LG Then the fol lowing

areequivalent

F stably o ccurs as the Gxedpoint data of a P Lproper Gaction on a

sphere

F stably occurs as the Gxedpoint data in a disk

satises BB

A nite group not of prime p ower order b elongs to exactly one of the following

six classes O

A G has a dihedral sub quotient of order n for a comp osite integer n

B G A and G has a comp osite order element conjugate to its inverse

C G AB G has a comp osite order element and the Sylow subgroups are

not normal in G

C G has a comp osite order element and the Sylow subgroup is normal in G

D G has no elements of comp osite order and the Sylow subgroups are not normal

in G

D G has no elements of comp osite order and the Sylow subgroup is normal in

ON FIXED POINT DATA OF SMOOTH ACTIONS ON SPHERES

Corollary B Let G be a nontrivial perfect group and F a closed manifold Then

F occurs as the Gxedpoint set of a P proper Gaction on a sphere if and only if

F occurs as the Gxedpoint set in a disk in other words

G A thereisnorestriction

G B c c KSpF TorK F

R F H

e g

G C r K F TorKOF

F C

G D TorKOF

Theorem C Let G be a nilpotent Oliver group and F a closed manifold Then

the fol lowing areequivalent

F occurs as the Gxedpoint set of a P proper Gaction on a sphere

is stably complex

F occurs as the Gxedpoint set of a Gaction on a disk

Our basic metho ds are

An extension of the metho d of equivariant bundles in O with mo dications

The equivariant thickening of P

The equivariant surgery results of M

References

BM Bak A and Morimoto M Equivariant surgery with midd le dimensional singular sets I

Forum Math

LM Laitinen E and Morimoto M Finite groups with smooth one xed point actions on

spheres Rep orts Dept Math Univ Helsinki no

M Morimoto M Equivariant surgery theory Construction of equivariant normal maps

Publ Res Inst Math Sci Kyoto Univ

M Equivariant surgery theory Deletinginserting theorem of xed point sets on

spheres Preprint April

MY Morimoto M and Yanagihara M On gap conditions of nite group actions Preprint

January

MY The gap conditions for S and GAP programsJFac EnvSci Tech Okayama

University

O Oliver R Fixedpoint sets of group actions on nite acyclic complexes Comment Math

Helvetici

O Oliver B Fixedpoint sets and tangent bund les of actions on disks and Euclidean spaces

to app ear in Top ology

P Pawalowski K Fixedpoint sets of smooth group actions on disks and Euclidean spaces

Top ology

Department of Environmental and Mathematical Sciences Okayama University

Okayama Japan

Email address morimotomathemsokayamauacjp

SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

Eiji Ogasa

Department of Mathematical Sciences

UniversityofTokyo

Komaba Tokyo

Japan

imunixccutokyoacjp

ogasamsredmsutokyoacjp

An oriented ordered mcomponent ndimensional link is a smo oth oriented

submanifold L fK K g of S which is the ordered disjoint union of m

manifolds each PL homeomorphic to the standard nsphere If m then L is

arelink called a knot Wesaythatmcomp onent ndimensional links L and L

concordant or linkcobordant if there is a smo oth oriented submanifold C fC

C g of S which meets the b oundary transversely in C is PL

homeomorphic to L and meets S fl g in L l

Wework in the smo oth category

b e spheres emb edded in the sphere S and intersect transversely and S Let S

Then the intersection C is adisjoint collection of circles Thus weobtaina pair of

in S and a pair of knots S links C in S

i i

Conversely let L L b e a pair of links and X X b e a pair of knots It is

natural to ask whether L L is obtained as the intersection of X and X

In this pap er wegive a complete answer to the ab ove question

Denition L L X X is called a quadruple of links if the following conditions

and hold

L K K is an oriented ordered m comp onent dimensional link

i i im i

m m X is an oriented knot i

DenitionA quadruple of links L L X X issaidtoberealizable if there exists

S satisfying the following conditions S a smo oth transverse immersion f S

f jS is a smo oth emb edding and denes the knot X i in S

denes the link L i For C f S f S the inverse image f C inS

This researchwas partially suppp orted by ResearchFellowships of the Promotion of Science for

Young Scientists

EIJI OGASA

Here the orientation of C is induced naturally from the preferred orientations

of S S and S and an arbitrary order is given to the comp onents of C

The following theorem characterizes the realizable quadruples of links

Theorem A quadruple of links L L X X is realizable if and only if L L

X X satises one of the fol lowing conditions i and ii

i Both L and L areproper links and

Arf L ArfL

ii Neither L nor L is proper and

lk K L K lk K L K mo d for all j

j j j j

Let f S S b e a smo oth transverse immersion with a connected selfintersec

tion C in S Then the inverse image f C inS is a knot or a comp onent link

For a similar realization problem wehave

Theorem

All component links arerealizable as above

Al l knots arerealizable as above

Remark By Theorem a quadruple of links L L X X withK b eing the

trivial knot and K b eing the trefoil knot is not realizable But by Theorem the

two comp onent split link of the trivial knot and the trefoil knot is realizable as the

selfintersection of an immersed sphere

We discuss the high dimensional analogue of x

Denition K K is called a pair of nknots if K and K are nknots

K K X X is called a quadruple of nknots and n knots or a quadruple

of n n knots if K and K constitute a pair of nknots K K and X and X

are dieomorphic to the standard n sphere

Denition A quadruple of n n knots K K X X issaidtobe realizable

n n

S if there exists a smo oth transverse immersion f S S satisfying the

following conditions

f jS denes X i

n n

The intersection f S f S is PL homeomorphic to the standard

sphere

f in S denes an nknot K i

A pair of nknots K K issaidtoberealizable if there is a quadruple of n n

knots K K X X which is realizable

The following theorem characterizes the realizable pairs of nknots

SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

Theorem A pair of nknots K K is realizable if and only if K K

satises the condition that

K K is arbitrary if n is even

Arf K ArfK if n m m m Z

K K if n m

There exists a mo d p erio dicity in dimension similar to the p erio dicity of high

dimensional knot cob ordism and surgery theory CS and L

Wehave the following results on the realization of a quadruple of n n knots

Theorem A quadruple of n n knots T K K X X is realizable if

K and K are slice

Kervaire proved that all even dimensional knots are slice Ke Hence wehave

Corollary If n is even an arbitrary quadruple of n nknots T K K X

X is realizable

In order to prove Theorem we intro duce a new knotting op eration for high

dimensional knots high dimensional passmoves The dimensional case of Denition

is discussed on p of K

Denition Let k knot K be dened by a smo oth emb edding g

k k

S where is PL homeomorphic to the standard k sphere k

k k

Let D fx x j x g and D fy y j y g Let

k k

x y

i i

k k

r fy y j y r g A D r fx x j x r g and D

k k

x y

i i

lo cal chart U ofS is called a passmovechart of K if it satises the following

conditions

k k k

R D D U

x y

k k k k k

gD D q f gD D g S U f

x y x y

k k

Let g S be an emb edding such that

k k

g jf g U g g jf g U gand

k k k

g U f gD D q

x y

k k k

gD D D f

x y y

k k

D D

x y

k k

f gD D g

x y

Let K be the k knot dened by g Then wesaythat K is obtained from K

U U U

bythe high dimensional passmove in U Wesaythatk knot K and K are

high dimensional passmove equivalent if there exist k knots K K

and passmovecharts U i qofK such that K K K K and

i q

K is obtained from K by the high dimensional passmovein U

i i i

variantand High dimensional passmoves have the following relation with the Arf in the signature of knots

EIJI OGASA

Theorem For simple k knots K and K the fol lowing two conditions

areequivalent k

K is passmove equivalent to K

Arf K ArfK when k is even

K and K satisfy the condition that

K K when k is odd

See L for simple knots

Theorem For k knots K and K the fol lowing two conditions areequiv

alent k

There exists a k knot K which is passmove equivalent to K and

cobordant to K

Arf K ArfK when k is even

K and K satisfy the condition that

K K when k is odd

The case k of Theorem follows from KK

We discuss the case when three spheres intersect in a sphere

Let F be closed surfaces i A surfaceF F F link is a

smo oth submanifold L K K K ofS where K is dieomorphic to F If

i i

F is orientable we assume that F is oriented and K is an oriented submanifold which

i i i

is orientation preserving dieomorphic to F If we call L a surfaceF knot

An F F link L K K is called a semiboundary link if

K H S K Zi j

i j

following S

An F F link L K K iscalledaboundary link if there exist Seifert hyp er

surfaces V for K i suchthat V V

i i

An F F link K K is called a split link if there exist B and B in S such

that B B and K B

Denition Let L K K L K K and L K K b e surface

links L L L is called a triple of surfacelinks if K is dieomorphic to K

ij ji

i j Note that the knot typ e of K is dierent from that of

Denition Let L K K L K K and L K K b e surface

links A triple of surfacelinks L L L is said to b e realizable if there exists a trans

verse immersion f S q S q S S such that f jS is an emb eddingi

and f f S f S f f S f S in S is L i j k

i j i i

Note If L L L is realizable then K are orientable and are given an orientation

naturally From now on we assume that when we say a triple of surfacelinks the

triple of surfacelinks consists of oriented surfacelinks

We state the main theorem

SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

Theorem Let L i besemiboundary surfacelinks Suppose the triple

of surfacelinks L L L is realizable Then we have the equality

L L L

where L is the SatoLevine invariant of L

i i

Refer to S for the SatoLevine invariant Since there exists a triple of surfacelinks

L L L suchthat L L and L R and S wehave

Corollary Not al l triples of oriente d surfacelinks arerealizable

Wehave sucient conditions for the realization

Theorem Let L i be split surfacelinks Then the triple of surface

links L L L is realizable

Theorem Suppose L are S S links If L areslice linksi then

i i

the triple of surfacelinks L L L is realizable

It is well known that there exists a slicelink which is neither a b oundary link nor

a ribb on link Hence wehave

Corollary There exists a realizable triple of surfacelinks L L L such that

neither L areboundary links and al l L are semiboundary links

i i

Besides the ab ove results weprove the following triple are realizable

Theorem There exists a realizable triple of surfacelinks L L L such that

neither L are semiboundary links

Here we state

Problem Suppose L L L Then is the triple of surfacelinks

L L L realizable

Using a result of O we can make another problem from Problem

Problem Is every triple of S S links realizable

Note By Theorem if the answer to Problem is negative then the answer to

an outstanding problem Is every S S link slice is no Refer CO to the

slice problem

An oriented ndimensional knot K is a smo oth oriented submanifold of R

R which is PL homeomorphic to the standard nsphere Wesay that nknots K

and K are equivalent if there exists an orientation preserving dieomorphism f

n n

R R R R such that f K K and f j K K is an orientation

n n

be the natural pro jection preserving dieomorphism Let R R R

map A subset P of R is called the projection of an nknot K if there exists an

EIJI OGASA

orientation preserving dieomorphism X K and a smo oth transverse immersion

X R such that j and j K X P The singular point

K K

set of the pro jection of an nknot is the set f x j K j j x g

K K

It is wellknown that the pro jection of any dimensional knot is the pro jection of

a knot equivalent to the trivial knot This fact is used in a way to dene the Jones

p olynomial and in another way to dene the ConwayAlexander p olynomial

We consider the following problem

Problem Let K be an nknot dieomorphic to the standard sphere Let P be

the pro jection of K Is P the pro jection of an nknot equivalent to the trivial knot

Furthermore supp ose that the singular p ointsetofP consists of double p oints

Is P the pro jection of an nknot equivalent to the trivial knot

The author proved that the answer to Problem in the case of n is negative

and hence that the answer to Problem in the case of n is also negative We

prove

Theorem Let n be any integer greater than two Thereexistsannknot K such

that the projection P has the fol lowing properties

P is not the projection of any knot equivalent to the trivial knot

The singular point set of P consists of double points

K is dieomorphic to the standard sphere

Note Problem in the case of n remains op en Dr Taniyama has informed

that the answer to Problem in the case of n is p ositive that the pro of is easy

but that he has not published the pro of

Wehave the following outstanding op en problems

Problem Classify nlinks up to link concordancefor n

Problem Is every even dimensional link slice

Problem Is every odd dimensional link concordant to a sublink of a homology

boundary link

The author has mo died Problem to formulate the following Problem We

n n

consider the case of comp onent links Let L K K S B be a

mlink m By Kervaires theorem in Ke there exist D i

m m m

m m

and S D K Then D emb edded in B such that D

i i

m m m m

denes D intersect mutually in general Furthermore D D in D

m link

m m

Problem Can we remove the above intersection D D by modifying

m m

embedding of D and D

If the answer to Problem is p ositive the answer to Problem is p ositive

Here wemake another problem from Problem

SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

m m

Problem What is obtained as a pair of m links D D in

m m m m m m

D D D in D bymodifying embedding of D and D

Wehave the following theorem which is an answer to Problem

Theorem For al l component mlink L K K m there exist

m m m m

D and D as above such that each of m links D D in

m m m m

D and D D in D is the trivial knot

Wehave the following Theorem Wesaythat ndimensional knots K and K

are linkconcordant or linkcobordant if there is a smo oth oriented submanifold C

of S which meets the b oundary transversely in C is PL homeomorphic to

L and meets S fl g in L l Then wecallC a concordancecylinder

of K and K

Theorem For al l component nlink L K K n there exist a

boundary link L K K satisfying that K is concordant to K and a concordance

K K C

such that each of n links C C in C and of cylinder

K K C

and C C in C is the trivial knot

When n is even Theorem is Theorem Because all even dimensional b ound

ary links are slice

By the following exciting theorem of Co chran and Orr when n is odd Theorem

is b est p ossible from a viewp oint

Theorem CO Not al l component odd dimensional links areconcordant to bound

ary links

n n n n

Let D D D be submanifolds of S dieomorphic to the ndisc such that

n n n n n n n n

Int D Int D for i j and D D D Then D D D

i j

is called an ndimensional curve in R The set of the constituent knots of an

n n

ndimensional curve in R is a set of three nknots in S which are made

n n n n n n

from D D D D and D D

The denitions in the case of the PL category are written in Y

Problem Takeany set of three nknots Is it the set of the constituent knots of an

ndimensional curve

In Y it is proved that if K K and K are ribb on nknots then the set

K K K is the set of the constituent knots of an ndimensional curve

We discuss the case of nonribb on knots The following theorems hold b oth in the

smo oth category and in the PL category

Wehave the following theorems

EIJI OGASA

Theorem Let n be any positive integer Let K and K be trivial knots

There exist an ndimensional curve in R and a nonribbon knot K such

that K K K is the set of the constituent knots of the ndimensional curve in

Furthermore wehave the following

Theorem Let m be any odd positive integer Let K and K be trivial knots

There exist mdimensional curves in R and a nonribbon and nonslice

knot K such that K K K is the set of the constituent knots of the mdimensional

curve in R

There exist mdimensional curves in R and a nonribbon and sliceknot

K such that K K K is the set of the constituent knots of the mdimensional

curve in R

Wehave the following

Theorem When n m m there exists a set of three nknots which

is never the set of the constituent knots of any curve

The ab ove problem remains op en

We use Theorem in xtogiveananswer to a problem of Fox

In F Fox submitted the following problem ab out links Here note that slice

link in the following problem is now called ordinary sense slice link and slice

link in the strong sense in the following problem is now called slice link byknot

theorists

Problem of F Find a necessary condition for L to b e a slice link a slice link

in the strong sense

Our purp ose is to give some answers to the former part of this problem The latter

half is not discussed here The latter half seems discussed much more often than the

former half See eg CO L etc

We review the denition of ordinary sense slice links and that of slice links which

wenowuse

We supp ose mcomp onent links are oriented and ordered

Let L K K beanmcomp onent link in S B L is called a slice

link which is a slice link in the strong sense in the sense of Fox if there exist

i j and D D B D i min B suchthatD discs D

j i i i i

D D inB denes L

Take a link L in S Take S and regard S as R R fg Regard the

sphere S as R fg in S L is called an ordinary sense slice linkwhichisa

slice link in the sense of Fox if there exists an emb edding f S R R suchthat

f is transverse to R fg and f S R fgin R fg denes L Supp ose f

denes a knot X Then L is called a crosssection of the knot X

SOME RESULTS ON KNOTS AND LINKS IN ALL DIMENSIONS

From nowonwe use the terms in the ordinary sense nowcurrent

Ordinary sense slice links have the following prop erties

Theorem Let L be a dimensional ordinary sense slice link Then the fol low

ings hold

L is a proper link

Arf L

Let K and K b e smo oth submanifolds of S dieomorphic to an ndimensional

closed smo oth manifold M The notion of cob ordism b etween K and K is dened

naturally A Seifert surface of K and a Seifert matrix of K are dened naturally

i i

The notion of matrix cob ordism b etween two Seifert matrices is dened naturally

It is also natural to ask the following problem

Problem Are K and K as ab ove cob ordant

The author thinks that there exists a kind of surgery exact sequence

The author obtained the following results

Theorem There exist a n dimensional closed orientedsmooth manifold

M and smooth submanifolds K and K of S dieomorphic to M such that

K and K are not cobordant and the Seifert matrices of K and K are matrix

cobordant

Theorem There exist a ndimensional closed orientedsmooth manifold M and

smooth submanifolds K and K of S dieomorphic to M such that K and K

arenotcobordant

In the b oth cases the obstructions live in certain homotopy groups

References

CS Capp ell S and Shaneson J L The codimension two placement problem and homology equiv

alent manifolds Ann of Math

CS Capp ell S and Shaneson J L Singular spaces characteristic classes and intersection ho

mology Ann of Math

CO Co chran T D and Orr K E Not al l links areconcordant to boundary linksAnnofMath

F Fox R H Some problems in knot theory Pro c Georgia Conf on the Top ology of

manifolds PrenticeHall pp

K Kauman L Formal knot theory Mathematical Notes Princeton UniversityPress

K Kauman L On Knots Ann of Maths Studies Princeton University Press

Ke Kervaire M Les noeudes de dimensions supereures BullSo cMathFrance

L Levine J Polynomial invariants of knots of codimension twoAnnMath

EIJI OGASA

L Levine J Knot cobordism in codimension two Comment Math Helv

L Levine J Link invariants via the etainvariant Comment Math Helv

Og Ogasa E On the intersection of spheres in a sphereIUniversityofTokyo preprint

Og Ogasa E On the intersection of spheres in a sphereIIUniversityofTokyo preprint

Og Ogasa E The intersection of more than three spheres in a sphere In preparation

O Orr K E New link invariants and applicationsComment Math Helv

R Rub erman D Concordance of links in S Contemp Math

S Sato N Cobordisms of semiboundary linksTop ology and its applications

Y Yasuhara A On higher dimensional curves Kob e J Math

CONTROLLED ALGEBRA AND TOPOLOGY

ERIK KJR PEDERSEN

Let R b e a ring and X X a pair of compact Hausdor spaces We assume

X X X is dense in X

Denition The continuously controlled category B X X R has ob jects A

fA g A a nitely generated free R mo dule satisfying that fxjA g is

x xX x x

lo cally nite in X

Given a subset U in X we dene AjU by

A if x U X

AjU

if x U X

X X R is an R mo dule morphism A B satisfy A morphism B

x y

ing a continuously controlled condition

z X U op en in X z U V op en in X z V

such that AjV AjU and AjX U AjX V

Clearly B X X R is an additive category with A B A B as direct

x x x

sum

If A is an ob ject of B X X R then fxjA g has no limit p ointinX all

limit p oints must b e in X We denote the set of limit p oints by supp A The

X X R on ob jects A with full sub category of B

supp A Z X

is denoted by B X X R Putting U B X X R and A B X X R

Z Z

this is a typical example of an Altered additive category U in the sense of Karoubi

The quotient category U A has the same ob jects as U buttwo morphisms are

identied if the dierence factors through an ob ject of A In the present example

this means two morphisms are identied if they agree on the ob ject restricted to a

X Z

neighborhood of X Z We denote U A in this case by B X X R Given

an ob ject A and a neighborhood W of X Z wehave A AjW in this category

If R is a ring with involution these categories b ecome additive categories with

involution in the sense of Ranicki It was proved in that

Theorem There is a bration of spectra

k h h

L A L U L U A

where k consists of projectives ie objects in the idempotent completion of Athat

become freeinU ie stably by adding objects in U bec ome isomorphic to an object

of U

The author was partially supp orted by NSF grant DMS

ERIK KJR PEDERSEN

Indication of proof Using the b ordism denition of Lsp ectra of Quinn and Ran

icki it is immediate that wehave a bration of sp ectra

h h h

L A L U L U A

h h

An elementinL U A the nth homotopy group of L U A is a pair of chain

complexes with b oundary in A and a quadratic Poincare structure The b oundary

is isomorphic to in U A since all Aob jects are isomorphic to in U A This

pro duces a map

h h

L U A L U A

which we ideally would like to be a homotopy equivalence Given a quadratic

Poincare complex in U A itiseasytoliftthechain complex to a chain complex

in U and to lift the quadratic structure but it is no longer a Poincare quadratic

structure Wemay use Prop to add a b oundary so that welifttoaPoincare

pair It follows that the b oundary is contractible in U A It turns out that a chain

complex in U is contractible in U A if and only if the chain complex is dominated

by a chain complex in A and such a chain complex is homotopy equivalent to

a chain complex in the idemp otent completion of A This is the reason for the

variation in the decorations in this theorem See for more details

Lemma If X X isacompact pair then

B X X R B CX X R

Proof The isomorphism is given by moving the mo dules A x X to point in

CX the same mo dule and if two are put the same place we take the direct sum

On morphisms the isomorphism is induced bytheidentitysowehavetoensurethe

continuously controlled condition is not violated Cho ose a metric on X so that all

distances are Given z X let y be a pointin X closest to z and send z to

dz yy Clearlyasz approaches the b oundary it is moved very little In the

other direction send t y to a p ointinB y t the ball with center y and radius

twhich is furthest awayfrom X Again moves b ecome small as t approaches

orequivalently as the p oint approaches X

Lemma L B X X R

Proof The rst denotes a point in X and the second that the sp ectrum is

contractible The pro of is an Eilenb erg swindle towards the p oint

Theorem The functor

B CY Y Z Y L

is a generalized homology theory on compact metric spaces

Proof Wehave a bration

h h h Y Z

L B CY Y Z L B CY Y Z L B CY Y Z

But an argument similar to the one used in Lemma shows

B CY Y Z B CZ Z Z

When everything is awayfrom Z it do es not matter if we collapse Z so wehave

Y Z YZ ZZ

Z B CY Z Y Z Z B CY Y

CONTROLLED ALGEBRA AND TOPOLOGY

but YZ ZZ is only one point from YZ so by Lemma the L sp ectrum is

homotopy equivalenttoL CY Z Y Z Z Finally Lemma shows that

B C YZ YZ Z B CY Z Y Z Z

and we are done

Consider a compact pair X Y so that X Y is a CW complex If we sub divide

so that cells in X Y b ecome small near Y the cellular chain complex C X Y Z

may b e thoughtofasachain complex in B X Y Zsimplybycho osing a p ointin

eachcell achoice whichisnoworse than the choice of the cellular structure If

wehave a strict map

f WY X Y

meaning f X Y W Y it is easy to see that given appropriate lo cal simple

connectedness conditions this map is a strict homotopy equivalence homotopies

through strict maps if and only if the induced map is a homotopy equivalence

of chain complexes in B X Y Z If the fundamental group of X Y is and

the universal cover satises the appropriate simply connectedness conditions strict

homotopyequivalence is measured bychain homotopy equivalences in B X Y Z

Wehave the ingredients of a surgery theory whichmaybedevelop ed along the lines

of with a surgery exact sequence

X Y

h h

B X Y Z S X Y F Top L

cc n

We will use this sequence to discuss a question originally considered in

nk k

Supp ose a nite group acts freely on S xing S a standard k

dimensional subsphere Wemay susp end this action to an action on S xing

S and the question arises whether a given action can b e desusp ended Notice this

question is only interesting in the top ological category In the PL or dierentiable

category it is clear that all such actions can b e maximally desusp ended by taking

alinkorby an equivariant smo oth normal bundle consideration

nk k

Denoting S S by X X is the homotopytyp e of a Swan complex

a nitely dominated space with universal cover homotopy equivalent to a sphere

nk k k k

The strict homotopytyp e of S S can b e seen to b e X S S

k k

and if wehave a strict homotopy equivalence from a manifold to X S S

it is easy to see that wemay complete to get a semifree action on a sphere xing

a standard subsphere This means that this kind of semifree action is classied by

the surgery exact sequence

k k

X S S

k k h

X F Top L B D S Z S

X S

n n

Now let C R R denote the sub category of B R R where the morphisms are

required to b e b ounded i e A B has to satisfy that there exists k k

n y

j k Radial shrinking denes a functor C R R so that if jx y

n n

B D S R and it is easy to see by the kind of arguments develop ed ab ove

that this functor induces isomorphism in Ltheory We get a map from the b ounded

ERIK KJR PEDERSEN

surgery exact sequence to the continuously controlled surgery exact sequence

X R

h k

S X F Top

L C R Z

k k

X S S

h k k h

L B D S Z S X F Top

which is an isomorphism on two out of three terms hence also on the structure set

This is useful b ecause we can not dene an op eration corresp onding to susp ension

of the action on the continuously controlled structure set An attempt would be

to cross with an op en interval but an op en interval would havetohave a sp ecic

cell structure to get a controlled algebraic Poincare structure on the interval but

then we would lose control along the susp ension lines In the b ounded context

susp ension corresp onds precisely to crossing with the reals and giving the reals a

b ounded triangulation we evidently have no trouble getting a map corresp onding

to crossing with R Since crossing with R kills torsion think of crossing with R

as crossing with S and pass to the universal cover we get a map from the h

structure set to the sstructure set The desusp ension problem is now determined

by the diagram

X R

h k h

S X F Top

L C R Z

X R

s k

S X F Top

L C R Z

with two out of three maps isomorphisms once again This shows wemay desusp end

if and only if the element in the structure set can b e thought of as a simple structure

ie if and only if an obstruction in

k k

WhC R Z K C R Z f g K Z

vanishes Since K Z for k this means we can always desusp end

untill we have a xed circle but then we encounter a p ossible obstruction The

computations in show these obstructions are realized

References

D R Anderson and E K Pedersen Semifree topological actions of nite groups on spheres

Math Ann

G Carlsson and E K Pedersen Control led algebra and the Novikov conjectures for K and

LtheoryTop ology

D Carter Lower K theory of nite groups Comm Alg

S C Ferry and E K Pedersen Epsilon surgery Theory Pro ceedings Ob erwolfachConfer

ence on the Novikov Conjectures Rigidity and Index Theorems Vol S C FerryJRosenberg

and A A Ranicki eds LMS Lecture Notes vol Cambridge University Press

nk k

I Hambleton and I Madsen Actions of nite groups on R with xed set R CanadJ

Math

M Karoubi Foncteur derives et Ktheorie Lecture Notes in Mathematics vol Springer

Verlag BerlinNew York

CONTROLLED ALGEBRA AND TOPOLOGY

A A Ranicki Algebraic Ltheory and Topological ManifoldsCambridge Tracts in Mathemat

ics vol Cambridge UniversityPressCambridge

Lower K and Ltheory Lond Math So c Lect Notes vol Cambridge Univ

Press Cambridge

Department of Mathematical Sciences SUNY at Binghamton Binghamton New York USA

PROBLEMS IN LOWDIMENSIONAL TOPOLOGY

Frank Quinn

Introduction

Fourdimensional top ology is in an unsettled state a great deal is known but

the largestscale patterns and basic unifying themes are not yet clear Kirbyhas

recently completed a massive review of lowdimensional problems Kirby and many

of the results assembled there are complicated and incomplete In this pap er the

fo cus is on a shorter list of to ol questions whose solution could unify and clarify

the situation However wewarn that these formulations are implicitly biased toward

p ositive solutions In other dimensions to ol questions are often directly settled one

way or the other and even a negative solution leads to a general conclusion eg

surgery obstructions Whitehead torsion characteristic classes etc In contrast

failures in dimension four tend to be indirect inferences and study of the failure

leads nowhere For instance the failure of the disk emb edding conjecture in the

smo oth category was inferred from Donaldsons nonexistence theorems for smo oth

manifolds And although some direct information ab out disks is nowavailable eg

Kr it do es not particularly illuminate the situation

Topics discussed are in section emb eddings of disks and spheres needed for

surgery and scob ordisms of manifolds Section describ es uniqueness questions

for these arising from the study of isotopies Section concerns handleb o dy struc

tures on manifolds Finally section poses a triangulation problem for certain

lowdimensional stratied spaces

This pap er was develop ed from a lecture given at the International Conference

on Surgery and Controlled Top ology held at Josai University in September I

would liketoexpressmy thanks to the organizers particularly Masayuki Yamasaki

and to Josai University for their great hospitality

disks and spheres in manifolds

The target results here are surgery and the scob ordism theorem In general

these are reduced via handleb o dy theory to questions ab out disks and spheres in

the middle dimension of the ambient manifold The to ol results hence the targets

are known in the top ological category for manifolds when the fundamental group

is small FQ FT but are unsettled in general

Two ndimensional submanifolds of a manifold of dimension n will usually in

tersect themselves and each other in isolated p oints The Whitney trick uses an

isotopy across an emb edded disk to simplify these intersections Roughly sp eak

ing this reduces the study of ndimensional emb eddings to emb eddings of disks

But this is not a reduction when the dimension is the disks themselves are

FRANK QUINN

middledimensional so trying to embed them encounters exactly the same prob

lems they are supp osed to solve This is the phenomenon that separates dimension

from others The central conjecture is that some emb eddings exist in spite of this

problem

Disk conjecture Suppose A is an immersion of a disk into a manifold

boundary going to boundary and thereisaframed immersed sphere B with trivial

algebraic selfintersection and algebraic intersection with A Then there is an

embedded disk with the same framedboundary as A

If this were true then the whole apparatus of highdimensional top ology would

apply in dimension There are very interesting generalizations which for example

ask ab out the minimal genus of an emb edded surface with a given b oundary or

Problem However the data in in a given homology class cf Kirby

is available in the Whitney disk applications so its inclusion reects the to ol

orientation of this pap er

The conjecture is very false for smo oth emb eddings since it would imply exis

tence and uniqueness results that are known to b e false Kirby Problems

It may b e true for top ological lo cally at emb eddings The current b est results

are by Freedman and Teichner FT FT In FT they show that the conjec

ture as stated holds if the fundamental group of the manifold has sub exp onential

growth while FT gives a technical but useful statementaboutemb eddings when

the manifold changes slightly We briey discuss the pro ofs

For surfaces in manifolds here is a corresp ondence b etween intersections and

fundamental group of the image adding an intersection point enlarges the fun

damental group of the image by one free generator if the image is connected

Freedmans work roughly gives a converse in order to removeintersections in M

it is sucient to kill the image of the fundamental group of the data in the fun

damental group of M More precisely if we add the hyp othesis that A B is a

single p oint and of the image A B is trivial in M then there is an emb edded

disk However applications of this dep end on the technology for reducing images in

fundamental groups Freedmans earlier work showed essentially howtochange

A and B so the fundamental group image b ecomes trivial under any M G

where G is p olynite or cyclic FT improves this to allow G of sub exp onential

growth Quite a lot of eort is required for this rather minute advance giving the

impression that we are near the limits of validity of the theorem In a nutshell

the new ingredient is the use of Milnor link homotopy Reduction of fundamental

group images is achieved by trading an intersection with a nontrivial lo op for a

great manyintersections with trivial or at least smaller lo ops The delicate p oint

is to avoid reintro ducing big lo ops through unwanted intersections The earlier

argument uses explicit moves The approach in FT uses a more ecient abstract

existence theorem The key is to think of a collection of disks as a nullhomotopyof

a link Selntersections are harmless while intersections b etween dierentcompo

nents are deadly Thus the nullhomotopies needed are exactly the ones studied by

y Milnor and existence of the desired disks can b e established using link homotop

invariants

While the conjecture is exp ected to b e false for arbitrary fundamental groups

no pro of is in sight Constructing an invariant to detect failure is a very delicate

PROBLEMS IN LOWDIMENSIONAL TOPOLOGY

limit problem The fundamental group of the image of the data can b e compressed

into arbitrarily farout terms in the lower central series of the fundamental group of

M If it could b e pushed into the intersection the general conjecture would follow

This is b ecause it is sucient to prove the conjecture for M with free fundamental

group eg a regular neighborhood of the data and the intersection of the lower

central series of a free group is trivial One approach is to develop a notion of

nesting of data so that the intersection of an innite nest gives something useful

Then in order for the theorem to fail there must b e data with no prop erly nested

sub data and mayb e this can b e detected

There is a mo dication of the conjecture in whichweallowtheambient manifold

to change by scob ordism This form implies that surgery works but not the s

cob ordism theorem FQ shows that if the fundamental group of the image of

the data of is trivial in the whole manifold then there is an emb edding up to

scob ordism This diers from the hyp othesis of the version ab oveinthat A B

is not required to b e one p oint just algebraically The improvementof FT is

roughly that innitesimal holes are allowed in the data A regular neighb orho o d of

the data gives a manifold with b oundary and carrying certain homology classes

In the regular neighb orho o d the homology class is represented by a sphere since a

sphere is given in the data The improvement relaxes this the homology class is

required to b e in a certain subgroup of H but not necessarily in the image of

Heuristically we can drill a hole in the sphere as long as it is small enough not to

move it to o far out of technically still in the term of Dwyers ltration on

The improved version has applications but again falls short of the full conjecture

Again it is a limit problem we can start with arbitrary data and drill very small

holes to get the image trivial in M The holes can be made small enough

that the resulting homology classes are in an arbitrarily farout term in the Dwyer

ltration but mayb e not in the innite intersection

There is still ro om for hop e that this form of the conjecture is true but it may

require a more elab orate construction or another innite pro cess A shell game

approach would begin with arbitrary data intro duce some S S summands

and use them as gently as p ossible to represent the original data as a trivial

submanifold with homology in Dwyers term The S S s are now messed up

and to repair this wewant to represent them also with trivial submanifolds with

ltration homology The new advantage is that the data is no longer random

given by an abstract existence theorem but is obtained from an emb edding by

carefully controlled damage done in the rst step An innite swindle would involve

intro ducing innitely many copies of S S and moving the damage down the

line The ob jectivewould b e to do this with control on sizes so the construction

will converge in an appropriate sense see BFMW The limit should b e an ANR

homology manifold but this can b e resolved to regain a top ological manifold Q

Uniqueness

The uniqueness question wewant to address is when are two homeomorphisms

of a manifold top ologically isotopic This is known for compact connected

manifolds Q but not for nontrivial groups even in the good class for surgery

FRANK QUINN

Neither is there a controlled version not even in the connected case The con

trolled version may b e more imp ortant than general fundamental groups since it

is the main missing ingredient in a general top ological isotopy extension theorem

for stratied sets Q

The study of isotopies is approached in two steps First determine if two home

omorphisms are concordant pseudoisotopic then see if the concordance is an iso

topy The rst step still works for manifolds since it uses dimensional surgery

The highdimensional approach to the second step HW reduces it to a to ol ques

tion However the uniqueness to ol question is not simply the uniqueness analog of

the existence question In applications Conjecture would b e used to nd Whit

ney disks to manipulate spheres The to ol question needed to analyse isotopies

directly concerns these Whitney disks

Conjecture Suppose A and B are framed embedded families of spheres

and V W are two sets of Whitney disks for eliminating AB intersections Each

set of Whitney disks reduces the intersections to make the families transverse the

spher es in A and B arepaired and the only intersections are a single point between

each pair Then the sets V W equivalent up to isotopy and disjoint replacement

Isotopic means there is an ambient isotopythat preserves the spheres A B

setwise and takes one set of disks to the other Note that A B must b e point

wise xed under such an isotopy Disjoint replacement means we declare two

sets to be equivalent if the only intersections are the endp oints in A B Ac

tually there are further restrictions on framings and homotopy classes related

to Hatchers secondary pseudoisotop y obstruction HW In practice these do not

b other us b ecause the work is done in a relative setting that enco des a vanishing

of the highdimensional obstruction we try to show that a dimensional concor

dance is an isotopy if and only if the pro duct with a disk is an isotopy In Q

this program is reduced to conjecture The conjecture itself is proved for simply

connected manifolds and A B each a single sphere

Consider the b oundary arcs of the disks V and W onA and B These t together

to form circles and arcs eachintersection p ointinA B is an endp oint of exactly

one arc in each of V A and W A unless it is one of the sp ecial intersections

left at the end of one of the deformations Thus there is exactly one arc on each

sphere The pro of of Q works on the arcs Focus on a single pair of spheres

The connectedness is used to merge the circles into the arc Intersections among

Whitney disks strung out along the arc are then pushed o the end of the arc

This makes the two sets of disks equivalent in the sense of and allows them to b e

cancelled from the picture Finitely many pairs can b e cancelled by iterating this

but this cannot b e done with control since each cancellation will greatly rearrange

the remaining spheres To get either nontrivial fundamental groups or control will

require dealing directly with the circles of Whitney arcs

dimensional handlebodies

Handleb o dy structures on manifolds corresp ond exactly to smo oth structures

The targets in studying handleb o dy structures are therefore the detection and ma

nipulation of smo oth structures However these are much more complicated than

in other dimensions and we are not yet in a p osition to identify to ol questions

PROBLEMS IN LOWDIMENSIONAL TOPOLOGY

that might unravel them Consequently the questions in this section suggest useful

directions rather than sp ecic problems

The rst problem concerns detection of structures The Donaldson and Seib erg

Witten invariants are dened using global dierential geometry But since a han

dleb o dy structure determines a smo oth structure these invariants are somehow

enco ded in the handle structure There can b e no direct top ological understanding

of these structures until we learn to deco de this

Problem Find a combinatorial lydened topological quantum eld theory

that detects exotic smooth structures

Threedimensional combinatorial eld theories were pioneered by Reshetikhin

and Turaev RT They attracted a lot of attention for a time but havenotyet led to

anything really substantial Fourdimensional attempts have not gotten anywhere

cf CKY The Donaldson and Seib ergWitten invariants do not satisfy the full set

of axioms currently used to dene a top ological quantum eld theory so there is

no guarantee that working in this framework will ever lead anywhere Nonetheless

this is curren tly our best hop e and a careful exploration of it will probably be

necessary b efore we can see something b etter

dimensional handleb o dies are describ ed by their attaching maps emb eddings

of circles and spheres in manifolds The dimension is low enough to draw

explicit pictures of many of these Kirby develop ed notations and a calculus

of such pictures for and handles cf HKK This approach has b een used

to analyse sp ecic manifolds a go o d example is Gompf s identication of some

homotopy spheres as standard Gf However this approach has b een limited even

in the study of examples b ecause

it only eectively tracks and handles and Gompf s example shows one

cannot aord to ignore handles

it is a nonalgorithmic art form that can hide mistakes from even skilled

practitioners and

there is no clue how the pictures relate to eective eg Donaldson and

Seib ergWitten invariants

The most interesting p ossibility for manipulating handleb o dies is suggested by

the work of Po enaru on the dimensional Poincare conjecture The following is

suggested as a test problem to develop the technique

Conjecture A dimensional smooth scobordism without hand les is a

product

Settling this would be an imp ortant advance but a lot of work remains be

fore it would have profound applications To some extentitwould show that the

real problem is getting rid of handles Kirby Problems It

have some application to this if we can arrange that some subset of the might

handles together with the handles forms an scob ordism then the dual han

dleb o dy structure has no handles and the conjecture would apply Replacing

these and handles with a pro duct structure gives a new handleb o dy without

handles The problem encountered here is control of the fundamental group of

the b oundary ab ove the handles The classical manipulations pro duce a homol

ogy scob ordism with Z co ecients but to get a genuine scob ordism we need

FRANK QUINN

for the new b oundary to have the same Thus to make progress wewould have

to understand the relationship b etween things like Seib ergWitten invariants and

restrictions on fundamental groups of b oundaries of subhandleb o dies

To analyse the conjecture consider the level between the and handles in

the scob ordism The attaching maps for the handles are spheres and the

dual spheres of the handles are circles The usual manipulations arrange the

algebraic intersection matrix b etween these to b e the identity In other dimensions

the next step is to realize this geometrically ndanisotopy of the circles so each

has exactly one point of intersection with the family of spheres But the usual

metho ds fail miserably in this dimension V Po enaru has attacked this problem in

the sp ecial case of I where is a homotopy ball P Gi The rough idea

is an innite pro cess in which one rep eatedly intro duces new cancelling pairs of

and handles then damages these in order to x the previous ones The limit has

an innite collections of circles and spheres with go o d intersections Unfortunately

this limit is a real mess top ologically in terms of things converging to each other

The goal is to see that by being incredibly clever and careful one can arrange

the spheres to converge to a singular lamination with control on the fundamental

groups of the complementary comp onents As an outline this makes a lot of sense

Unfortunately Po enarus manuscript is extremely long and complicated and as a

result of manyyears of work without feedback from the rest of the mathematical

community is quite idiosyncratic It would probably takeyears of eort to extract

clues from this on how to deal with the dicult parts

Stratified spaces

A class of stratied spaces with a relatively weak relationship b etween the strata

has emerged as the prop er setting for purely top ological stratied questions see

eg Q W The analysis of these sets to obtain results like isotopy extension

theorems uses a great deal of handleb o dy theory etc so often requires the as

sumption that all strata have dimension or greater This restriction is acceptable

in some applications for example in group actions but not in others likesmooth

singularity theory algebraic varieties and limit problems in dierential geometry

The suggestion here is that many of the lowdimensional issues can b e reduced to

much easier PL and dierential top ology The conjecture as formulated is a

to ol question for applications of stratied sets After the statementwe discuss its

dissection into top ological to ol questions

Conjecture A threedimensional homotopical ly stratied space with mani

fold strata is triangulable A dimensional spaceofthistype is triangulable in the

complement of a discrete set of points

As stated this implies the dimensional Poincare conjecture Toavoid this as

sume either that there are no fake balls b elow a certain diameter or change the

statement to obtained from a p olyhedron by replacing sequences of balls converg

ing to the skeleton by fake balls The Hauptvermutung for dimensional

p olyhedra Papa asserts that homeomorphisms are isotopic to PL homeomor

phisms This reduces the dimensional version to showing that stratied spaces

are lo cally triangulable The skeleton and its complement are b oth triangulable

so the problem concerns how the dimensional part approaches neighb orho o ds of

PROBLEMS IN LOWDIMENSIONAL TOPOLOGY

points in the skeleton Consider a manifold p oint in the skeleton a neighborhood

in the skeleton is isomorphic with R for n or Near this the stratum

lo oks lo cally homotopically like a bration over R with b er a Poincare space of

dimension n We can reduce to the case where the b er is connected by

considering comp onents of the stratum one at a time If n then the b er

is a p oint and the union of the two strata is a homology manifold with R as

b oundary Thus the question is this union a manifold or equivalently is the R

collared in the union If n then the b er is S and the union gives an arc

homotopically tamely emb edded in the interior of a homology manifold Is it

lo cally at Finally if n then the b er is a surface dimensional Poincare

spaces are surfaces EL This is an end problem if a manifold has a tame end

homotopic to S R S a surface is the end collared Answers to these are probably

known The next step is to consider a p oint in the closure of strata of three dier

ent dimensions There are three cases and Again each

case can b e describ ed quite explicitly and should either b e known or accessible to

standard manifold techniques

Now consider dimensional spaces manifolds are triangulable in the com

plement of a discrete set so again the question concerns neighb orho o ds of the

skeleton In dimension homeomorphism generally do es not imply PL isomor

phism so this do es not immediately reduce to a lo cal question However the ob

jective is to construct bundlelike structures in a neighb orho o d of the skeleton and

homeomorphism of total spaces of bundles in most cases will imply isomorphism of

bundles So the question might b e lo calized in this way or just approached glob

ally using relativeversions of the lo cal questions As ab ovewe start with manifold

points in the skeleton If the p oint has a or disk neighb orho o d then the question

reduces to lo cal atness of b oundaries or manifolds in a homology manifold see

Q FQ A If the p oint has a disk neighb orho o d then a neighb orho o d lo oks

homotopically like the mapping cylinder of a surface bundle over R This leads

to the question is it homeomorphic to such a mapping cylinder If the surface

fundamental group has sub exp onential growth then this probably can be settled

by current techniques but the general case may have to wait on solution of the

conjectures of section Finally neighborhoods of isolated points in the skeleton

corresp ond exactly to tame ends of manifolds Some of these are knownnottobe

triangulable so these would have to b e among the p oints that the statement allows

to b e deleted From here the analysis progresses to p oints in the closure of strata

of three or four dierent dimensions Again there are a small numb er of cases each

of which has a detailed lo cal homotopical description

References

BFMW J Bryant S FerryWMioandSWeinb erger Topology of homology manifolds Ann

Math

EL B Eckmann and P Linnell Poincare duality groups of dimension II Comment

Math Helv

Kirby R Kirby Problems in lowdimensional topology

Kr P B Kronheimer An obstruction to removing intersection points in immersedsur

faces preprint

FQ M Freedman and F Quinn Topology of manifolds Princeton University Press

FRANK QUINN

FT M Freedman and PTeichner manifold topology I Subexponential groupsInvent

Math

FT manifold topology II Dwyers ltration and surgery kernelsInvent Math

Gi D Gabai Valentin Poenarus program for the Poincareconjecture Geometry top ology

and physics for Raoul Bott ST Yau ed International Press pp

Gf R Gompf Kil ling the AkbulutKirby sphere with relevancetotheAndrewsCurtis

and Schoenies problemsTop ology

HKK J Harer A Kas and R Kirby Hand lebody decompositions of complex surfaces Mem

oirs of the Amer Math So c

HW A Hatcher and J Wagoner Pseudoisotopy of compact manifolds Asterisque

So c Math France

Papa C Papakyriakop oulos A new proof of the invariance of the homology groups of a

ece complex Bull So c Math Gr

P V Po enaru The strange compactication theorem unpublished manuscript

Q F Quinn Ends of maps III Dimensions and J Di Geometry

Isotopy of manifolds J Di Geometry Q

Ends of maps IV Pseudoisotopy Am J Math Q

Q Homotopical ly stratiedsets J Amer Math So c

W S Weinb erger The topological classication of stratiedspacesUniversity of Chicago

Press

Department of Mathematics Virginia Polytech Institute State University

Blacksburg VA USA

Email address quinnmathvtedu

SLIDES ON CHAIN DUALITY

ANDREW RANICKI

Abstract The texts of slides on the applications of chain duality to the ho

mological analysis of the singularities of Poincare complexes the double p oints of

maps of manifolds and to surgery theory

Introduction

Poincare duality

H M H M

is the basic algebraic prop ertyofanndimensional manifold M

Achain complex C with ndimensional Poincare duality

H C H C

is an algebraic mo del for an ndimensional manifold generalizing the intersection

form

Spaces with Poincare duality such as manifolds determine Poincare dual

ity chain complexes in additive categories with chain duality giving rise to

interesting invariants old and new

What is chain duality

A additive category

B A additive category of nite chain complexes in A

Acontravariant additive functor T A B A extends to

T B A B A C T C

by the total double complex

T C T C

n p q

pq n

T e on A is a contravariant additive functor Denition A chain duality

T A B A together with a natural transformation e T A B A

such that for each ob ject A in A

eT A TeA T A T A

eA T A A is a chain equivalence

The lecture at the conference on Surgery and Geometric Top ology Josai University Japan

on Septemb er used slides

ANDREW RANICKI

Properties of chain duality

The dual of an ob ject A is a chain complex T A

The dual of a chain complex C is a chain complex T C

Motivated byVerdier duality in sheaf theory

ARanicki Algebraic Ltheory and top ological manifolds

Tracts in Mathematics Cambridge

Involutions

An involution T eon an additive category A is a chain duality such that

T A is a dimensional chain complex ob ject for each ob ject A in A

with eAT A A an isomorphism

ExampleAninvolution R R r r on a ring R determines the involution

T e on the additive category A R of fg free left R mo dules

T A Hom A R

R T A T A rf x f xr

f x eA A T A x f

Manifolds and homeomorphisms up to homotopy

Traditional questions of surgery theory

Is a space with Poincare duality homotopy equivalenttoa

manifold

Is a homotopy equivalence of manifolds homotopic to a

homeomorphism

Answered for dimensions by surgery exact sequence in terms of the

assembly map

A H X L Z L Z X

Ltheory of additive categories with involution suces for surgery groups

L Z X

Need chain duality for the generalized homology groups H X L Z and A

Manifolds and homeomorphisms

Will use chain duality to answer questions of the typ e

Is a space with Poincare duality a manifold

Is a homotopy equivalence of manifolds a homeomorphism

Controlled topology

Controlled top ology ChapmanFerryQuinn considers

the approximation of manifolds byPoincare complexes

the appro ximation of homeomorphisms of manifolds by homotopy equiv

alences

Philosophy of controlled top ology with control map X X

A Poincare complex X is a homology manifold if and only if it is an

controlled Poincare complex for all

A map of homology manifolds f M X has contractible p ointinverses

if and only if it is an controlled homotopy equivalence for all

SLIDES ON CHAIN DUALITY

Simplicial complexes

In dealing with applications of chain duality to top ology will only work with

connected nite simplicial complexes and oriented p olyhedral homology

manifolds and Poincare complexes

Can also work with sets and top ological spaces using the metho ds of

Forum Math MWeiss Visible Ltheory

SHutt Poincare sheaves on top ological spaces Trans AMS

Simplicial control

Additive category A ZXofX controlled Zmo dules for a simplicial complex

ARanicki and MWeiss Chain complexes and assembly Math Z

Will use chain dualityonA ZX to obtain homological obstructions for de

ciding

Is a simplicial Poincare complex X a homology manifold

Singularities

Do es a degree map f M X of p olyhedral homology manifolds have

acyclic p ointinverses Double p oints

Acyclic p ointinverses H f x is analogue of homeomorphism in the

world of homology

The X controlled Zmodule category A ZX

X simplicial complex

is a nitely generated free Zmo dule A with direct sum A ZXmo dule

decomp osition

A A

AZXmo dule morphism f A B is a Zmo dule morphism suchthat

f A B

Prop osition AZXmo dule chain map f C D is a chain equivalence

if and only if the Zmo dule chain maps

f C D X

are chain equivalences

Functorial formulation

Regard simplicial complex X as the category with

ob jects simplexes X

morphisms face inclusions

determines a contravariant functor AZXmo dule A A

AX A Zffg free ab elian groupsg A A

ANDREW RANICKI

The ZXmo dule category A ZX is a full sub category of the category of

contravariant functors X A Z

Dual cells

The barycentric sub division X of X is the simplicial complex with one n

simplex b b b for each sequence of simplexes in X

of a simplex X is the contractible sub complex The dual cell

D X fb b b j g X

with b oundary

DX fb b b j g D X

Intro duced byPoincaretoprove duality

A simplicial map f M X has acyclic p ointinverses if and only if

f j H f D X H D X X

Where do ZXmodule chain complexes come from

For any simplicial map f M X the simplicial chain complex M isa

ZXmo dule chain complex

M f D Xf DX

with a degreewise direct sum decomp osition

M M f D X

The simplicial co chain complex X is a ZX mo dule chain complex

with

Z if r j j

otherwise

The ZXmodule chain duality

Prop osition The additive category A ZXofZXmo dules has a chain

dualityT ewith

T A Hom Hom X A Z

j j

TA A

Hom A Z if r j j

T A

if r j j

T C Hom C X Hom C Z

Z Z Z

T X X

T C T C n Terminology

SLIDES ON CHAIN DUALITY

Products

The pro duct of ZXmo dules A B is the ZXmo dule

A B A B A B

Z Z

A B A B

C X C

ZX ZX

T C D Hom C D

ZX ZX

For simplicial maps f M X g N X

M N f g

ZX ZX

T M T N M N M N nf g

Z X

Cap product

The AlexanderWhitney diagonal chain approximation

X X X

xb xb xb xb xb xb

n i i n

is the comp osite of a chain equivalence

X X X

ZX ZX

and the inclusion

X X X X

Homology classes X H X are in oneone corresp ondence with the chain

homotopy classes of ZXmo dule chain maps

X X X

Homology manifolds

manifold Denition A simplicial complex X is an ndimensional homology

Z if n

H X X nb X

otherwise

Prop osition A simplicial complex X is an ndimensional homology manifold

if and only if there exists a homology class X H X such that the cap

pro duct

X X X

is a ZXmo dule chain equivalence

Pro of For any simplicial complex X

H X X nb H D XDX

j j

Z if n

D X X H

otherwise

ANDREW RANICKI

Poincare complexes

Denition An ndimensional Poincare complex X is a simplicial complex

with a homology class X H X such that

X H X H X

Poincare duality theorem An ndimensional homology manifold X is an

ndimensional Poincare complex

Pro of AZXmo dule chain equivalence

X X X

is a Zmo dule chain equivalence

There is also a Z X version

McCrorys Theorem

X ndimensional Poincar e complex

X X isandimensional Poincare complex

Let V H X X bethePoincaredualof X H X X

Exact sequence

n n n

H X X X X n H X X H X X n

X X

Theorem McCrory X is an ndimensional homology manifold if and

only if V has image H X X n

Acharacterization of homology manifolds J Lond Math So c

Chain duality proof of McCrorys Theorem

n n

V has image H X X n if and only if there exists U H X

X X X n with image V

U isachain homotopy class of ZXmo dule chain maps X X

since

H X X X X n H T X T X

X n

H Hom X X

U is a chain homotopyinverse of

X X X

with

U H Hom X X H X

T TU TUT T U

SLIDES ON CHAIN DUALITY

The homology tangent bundle

The tangent bundle of a manifold X is the normal bundle of the diagonal

emb edding

X X X x x x

The homology tangent bundle of an ndimensional homology manifold X

is the bration

X X nfg X X X X n X

with X X X x y x

Thom space of

T X X X X n

X X

of Thom class

n n

U H T H X X X X n

X X

has image V H X X

Euler

of a simplicial complex X is The Euler characteristic

dim H X R Z X

R r

For an ndimensional Poincare complex X

X V H X Z

The Euler class of nplane bundle over X

n n

e U im H T H X

Reformulation of McCrorys Theorem

an ndimensional Poincare complex X is a homology manifold if and only if

n n

V H X X is the image of Thom class U H T in whichcase

X e H X Z

Degree maps

Amapf M X of ndimensional Poincare complexes has degree if

f M X H X

A homology equivalence has degree

Zmo dule chain map of a degree map f M X The Umkehr

n n

f X X M M

is such that ff X X

A degree map f is a homology equivalence if and only if

f f M M

if and only if

f f X M H M M

ANDREW RANICKI

The double point set

Do es a degree map of ndimensional homology manifolds f M X have

acyclic p ointinverses

Obstruction in homology of double p ointset

f f fx y M M j f xf y X g

Dene maps

i M f f a a a

j f f X x y f xf y

such that f ji M X

The Umkehr map

j H X H X X X X n

n X

H M M M M nf f

H f f Lefschetz duality

n X

is such that j j

Lefschetz

Lefschetz duality If W is an mdimensional homology manifold and A W

is a sub complex then

H WWnA H A

For any regular neighb ourho o d V V of A in W there are dened Pro of

isomorphisms

H WWnA H WWnV homotopyinvariance

H W W nV collaring

H V V excision

H V PoincareLefschetz duality

H A homotopyinvariance

Alexander duality is the sp ecial case W S

Acyclic Point Inverse Theorem

Theorem A degree map f M X of ndimensional homology

manifolds has acyclic p ointinverses if and only if

i M j X H f f

n X

Equivalent conditions

H f f i H M

n X n

H f f i H M

H f f n

X M

SLIDES ON CHAIN DUALITY

Conditions satised if f M X is injective with

f f

X M

In general i j f and i M j X

Proof of Theorem PartI

A simplicial map f M X has acyclic point inverses if and only if f

M X isaZXmo dule chain equivalence

For degree map f M X of ndimensional homology manifolds dene

the Umkehr ZXmo dule chain map

n n

f X X M M

f is a chain homotopyrightinverse for f

ff X X

f is also a chain homotopy left inverse for f if and only if

f f H Hom M M

Proof of Theorem PartII

Use the ZXPoincar e duality

M M

and the prop erties of chain dualityin A ZXtoidentify

i M f f j X H Hom M M

H Hom M M

H M M

H f f

n X

Cohomology version of Theorem

Theorem A degree map f M X of ndimensional homology

manifolds has acyclic p ointinverses if and only if the Thom classes U

n n

H M M M M n U H X X X X n have the same

M X X

image in H M M M M nf f

Same pro of as homology version after Lefschetz duality identications

U M H M M M M n H M

M M n

U X H X X X X n H X

X X n

H M M M M nf f H f f

X n X

ANDREW RANICKI

The double point obstruction

The double p oint obstruction of a degree map f M X of homology

manifolds

i M j X H f f

n X

is if and only if f has acyclic p ointinverses

The obstruction has image

M X H M Z

If f is covered by a map of homology tangent bundles

b M M M M n X X X X n

M X

then

U b U H M M M M n

M X M

the double p oint obstruction is and f has acyclic p ointinverses

Normal maps

A degree map f M X of ndimensional homology manifolds is normal

if it is covered by a map b of the stable tangent bundles

M X

The stable map of Thom spaces

T b T T

M X

induces a map in cohomology

n n

T b H T H X X X X n

X X

n n

H T H M M M M n

M M

which sends the Thom class U to U

X M

Howev er Theorem may not apply to a normal map f bM X since in

general

f f inclusion T b H T

H M M M M nf f

dual of i j f

The surgery obstruction

The Wall surgery obstruction of a degree normal map f b M X of

ndimensional homology manifolds

X f b L Z

is if and for n only if f b is normal b ordant to a homotopy equiva

lence

A degree map f M X with acyclic p ointinverses is a normal map with

zero surgery obstruction

What is the relationship b etween the double p oint obstruction of a degree

normal map f bM X and the surgery obstruction

Use chain level surgery obstruction theory

ARanicki The algebraic theory of surgery Proc Lond Math So c

SLIDES ON CHAIN DUALITY

Quadratic Poincare complexes

The simplyconnected surgery obstruction f b L Z is the cob ordism

class of the ndimensional quadratic Poincare complex

C C f e e

where

e M C f is the inclusion in the algebraic mapping cone of the

Zmo dule chain map f X M

the quadratic structure is the image of

H E M M H W M M

b n n Z

E S a contractible space with a free action

W E

There is also a Z X version

The double point and surgery obstructions PartI

For anydegreemap f M X of ndimensional homology manifolds the

comp osite of

i f j H X H f f

and H f f H M M is

f f f H X H M M

For a degree normal map f bM X

H f f H M M

n X n

sends the double p oint obstruction i M j X to

T M f f X H M M

b n

T ifandonlyif f is a homology equivalence

The double point and surgery obstructions PartII

A degree normal map f bM X of ndimensional homology manifolds

determines the X controlled quadratic structure

H E f f

bX n X

H W M M

Z ZX

has images

the quadratic structure

M M H E

bX b n

the double p oint obstruction

T i M j X H f f

bX n X

ANDREW RANICKI

The normal invariant

The X controlled quadratic Poincare cob ordism class

f bC f e e L A ZX H X L Z

bX n n

is the normal invariant of an ndimensional degree normal map f bM

f b if and for n only if f b is normal b ordanttoa mapwith

acyclic p ointinverses

The nonsimplyconnected surgery obstruction of f b is the assembly of the

normal invariant

f bA f b L Z X

Hom and Derived Hom

For ZXmo dules A B the additive group Hom A B do es not havea

natural ZXmo dule structure but the chain duality determines a natural

ZXmo dule resolution

Derived Hom of ZXmo dule chain complexes C D

RHom C D T C D

ZX ZX

Adjoint prop erties

RHom C D Hom C D

ZX ZX

RHom T C D C D

ZX ZX ZX

D X is the dualizing complex for chain duality

T C RHom C X

ZX ZX

as for Verdier duality in sheaf theory

When is a Poincare complex homotopy equivalent to a manifold

Every ndimensional top ological manifold is homotopy equivalent to an n

dimensional Poincare complex

Is every ndimensional Poincare complex homotopy equivalenttoanndimensional

top ological manifold

From nowonn

BrowderNovikovSullivanWall obstruction theory has b een reformulated in

terms of chain duality

the total surgery obstruction

BrowderNovikovSullivanWall theory

An ndimensional Poincare complex X is homotopy equivalenttoanndimensional

top ological manifold if and only if

the Spivak normal bration of X admits a top ological reduction

there exists a reduction such that the corresp onding normal map f b

M X has surgery obstruction

f b L Z X

SLIDES ON CHAIN DUALITY

Algebraic Poincare cobordism

ring with involution

L Wall surgery obstruction group

the cob ordism group of ndimensional quadratic Poincare

complexes over

ndimensional fg free mo dule chain complexes C with

H C H C

uses ordinary duality

C Hom C

Assembly

X connected simplicial complex

X universal cover

p X X covering pro jection

Assembly functor

A A ZXfZXmo dulesg A Z X fZ X mo dulesg

P P

M M M X M pe

e X

The assembly AT M of dual ZXmo dule chain complex

T M Hom M X

is chain equivalent to dual Z X mo dule

e e

M X Hom M X Z X

Z X

The algebraic surgery exact sequence

For any simplicial complex X exact sequence

H X L Z L Z X S X H X L Z

n n n n

with

A assembly

L Z the connective simplyconnected surgery sp ectrum

L Z L Z

H X L Z generalized homology group

cob ordism group of ndimensional quadratic PoincareZXmo dule com

plexes C T C

uses chain duality

T C RHom C X

ZX ZX

The structure group

X simplicial complex

S X structure group

S X cob ordism group of

n dimensional quadratic PoincareZXmo dule complexes

with contractible Z X mo dule assembly

ANDREW RANICKI

Local and global Poincare duality

X ndimensional Poincare complex

The cap pro duct X X X

is a ZXmo dule chain map

assembles to Z X mo dule chain equivalence

e e

X X X

The algebraic mapping cone

C C X X X

is an n dimensional quadratic PoincareZXmo dule

complex

with contractible Z X assembly

X is a homology manifold if and only if C is ZXcontractible

The total surgery obstruction

X ndimensional Poincare complex

of X is the cob ordism class The total surgery obstruction

sX C X S X

Theorem X is homotopy equivalenttoanndimensional top ological man

ifold if and only if sX S X

Theorem Ahomotopy equivalence f M N of ndimensional top olog

ical manifolds has a total surgery obstruction sf S N suchthatf is

homotopic to a homeomorphism if and only if sf

Should also consider Whitehead torsion

University of Edinburgh Edinburgh Scotland UK

Email address aarmathsedacuk

Controlled LTheory

Preliminary announcement

A Ranicki and M Yamasaki

Intro duction

This is a preliminary announcement of a controlled algebraic surgery theoryof

the typ e rst prop osed by Quinn We dene and study the controlled Lgroups

L X p extending to Ltheory the controlled K theory of Ranicki and Yamasaki

n X

The most immediate application of the algebra to controlled geometric surgery

is the controlled surgery obstruction a normal map f b K L from a closed

ndimensional manifold to a controlled Poincare complex determines an element

f b L X

n X

a n The construction in Ranicki and Yamasaki can be used to pro duce

dimensional quadratic Poincare structure on an n dimensional chain complex

There is a chain equivalence from this to an ndimensional chain complex with a

ndimensional quadratic Poincare structure and f b is the cob ordism class of

this complex in L X A relative construction shows that if f bcanbe

n X

made into a controlled homotopy equivalence by controlled surgery then

f b L X

n X

Converselyif n andf bissuch that

f b L X

n X

then f b can be made into an controlled homotopy equivalence by controlled

surgery where C for a certain constant C that dep ends on n Pro ofs

of dicult results and the applications of the algebra to top ology are deferred to the

nal account

The algebraic prop erties required to obtain these applications include the con

trolled Ltheory analogues of the homology exact sequence of a pair and

the MayerVietoris sequence

The limit of the controlled Lgroups

L X lim lim imfL X L X g

X n X n X

is the obstruction group for controlled surgery to controlled homotopy equivalence

for all

A RANICKI AND M YAMASAKI

Theorem Fix a compact p olyhedron X andaninteger n There exist

numbers and suchthat

L X imfL X L X g

X n X n X

for every and every

Throughout this pap er all the mo dules are assumed to b e nitely generated unless

otherwise stated explicitly But note that all the denitions and the constructions are

valid also for p ossiblyinnitelygenerated mo dules and chain complexes Actually we

heavily use nite dimensional but innitely generated chain complexes in the later

R comes into the game part of the pap er That is where the b oundedcontrol over

So we rst pretend that everything is nitely generated and later we intro duce a

p ossiblyinnitelygenerated analogue without any details

Epsiloncontrolled Lgroups

In this section weintro duce controlled Lgroups L X p andL X Y p for

n X n X

p M X Y X n These are dened using geometric mo dule chain

complexes with quadratic Poincare structures whichwere discussed in Yamasaki

We use the convention in Ranicki and Yamasaki for radii of geometric mor

phisms etc The dual of a geometric mo dule is the geometric mo dule itself and the

dual of a geometric morphism is dened byreversing the orientation of paths Note

that if f has radius then so do es its dual f and that f g implies f g by

our convention For a geometric mo dule chain complex C its dual C is dened

using the sign convention used in Ranicki

Forasubset S of a metric space X S will denote the closed neighborhood of

S in X when When S will denote the set X X S

Let C b e a free chain complex on p M X An ndimensional quadratic

structure on C is a collection f js g of geometric morphisms

n r s

C C C r Z

s nr s r

of radius suchthat

r ns s nr s

d d T C C

s s s s r

for s An ndimensional free chain complex C on p equipp ed with an n

dimensional quadratic complex on dimensional quadratic structure is called an n

p Here a complex C is ndimensional if C for iand i n

X i

CONTROLLED LTHEORY

Next let f C D be achain map b etween free chain complexes on p An

n dimensional quadratic structure on f is a collection f js g

s s

of geometric morphisms

nr s nr s

D D C C r Z

s r s r

of radius such that the following holds in addition to

r n s s n

d d T f f

s s s s s

nr s

D D s

An chain map f C D between an ndimensional free chain complex C on p

and an n dimensional free chain complex D on p equipp ed with an n

dimensional quadratic structure is called an n dimensional quadratic pair on

p Obviously its b oundary C isanndimensional quadratic complex on p

X X

cob ordism of ndimensional quadratic structures on C and on C An

is an n dimensional quadratic structure on some chain map

C C D An cob ordism of ndimensional quadratic complexes C C

on p is an n dimensional quadratic pair on p

X X

f f C C D

with b oundary C C The union of adjoining cob ordisms are dened

using the formula in Chapter of Ranicki The union of adjoining cob ordisms

is a cob ordism

C and C will denote the susp ension and the desusp ension of C resp ectively

and C f will denote the algebraic mapping cone of a chain map f

Denition Let W beasubsetofX An ndimensional quadratic structure on

C is Poincareover W if the algebraic mapping cone of the duality chain map

D T C C

is contractible over W A quadratic complex C is Poincareover W if is

eover W Similarlyann dimensional quadratic structure Poincar

on f C D is Poincareover W if the algebraic mapping cone of the duality

chain map

D T f T C f D

is contractible over W or equivalently the algebraic mapping cone of the chain

map

D D Cf D C

r r r

T f

is contractible over W and is Poincareover W A quadratic pair f

is Poincareover W if is Poincareover W We will also use the notation

D T although it do es not dene a chain map from D to D in

general

A RANICKI AND M YAMASAKI

Denition A p ositive geometric chain complex C C for i is

connected if there exists a morphism h C C suchthatdh

A chain map f C D of p ositivechain complexes is connected if C f is

connected

A quadratic complex C is connected if D is connected

A quadratic pair f C D is connected if D and D are

connected

Nowwe dene the controlled Lgroups Let Y b e a subset of X

Denition For n and L X Y p is dened to be the equivalence

n X

classes of ndimensional connected quadratic complexes on p that are Poincare

over X Y The equivalence relation is generated by connected cob ordisms that

are Poincareover X Y For Y write

L X p L X p

X n X n

Remarks We use only ndimensional complexes and not the complexes chain

equivalentto ndimensional ones in order to makesurewehave size control on some

constructions

The connectedness condition is automatic for complexes that are Poincareover

X Connectedness condition is used to insure that the b oundary C C D is

chain equivalent to a p ositive one There is a quadratic structure for C so that

C isPoincareRanicki

Using lo callynitely generated chain complexes on M one can similarly dene

controlled lo callynite Lgroups L sections X Y p All the results in

are valid for lo callynite Lgroups

Prop osition The direct sum

C C C C

induces an ab elian group structure on L X Y p Furthermore if

n X

C C L X Y p

n X

then there is a connected cob ordism b etween C and C that is

Poincareover X Y

Next we study the functoriality A map b etw een control maps p M X and

p N Y means a pair of continuous maps f M N f X Y which makes

the following diagram commute

N M

p p

X Y

CONTROLLED LTHEORY

For example given a control map p N Y and a subset X Y let us denote the

control map p jp X p X X by p M X Then the inclusion maps

Y X

Y Y

j M N j X Y form a map form p to p

X Y

Epsilon controlled Lgroups are functorial with resp ect to maps and relaxation

of control in the following sense

Prop osition Let F f f b e a map from p M X to p N Y and

X Y

supp ose that f is Lipschitz continuous with Lipschitz constant ie there exists a

constant suchthat

df x f x dx x x x X

Then F induces a homomorphism

F L X X p L Y Y p

n X n Y

if and f X Y If two maps F f f and G g g are homotopic

through maps H h h suchthateach h is Lipschitz continuous with Lipschitz

t t t t

constant and h X Y then the following two comp ositions are

the same

L X X p L Y Y p L Y Y p

n X n Y n Y

L X X p L Y Y p L Y Y p

n X n Y n Y

Pro of The direct image construction for geometric mo dules and morphisms p

can b e used to dene the direct images f C of quadratic complexes and the direct

images of cob ordism And this induces the desired F The rst part is obvious For

the second part split the homotopy in small pieces to construct small cob ordisms

The size of the cob ordism maybeslightly bigger than the size of the ob ject itself

Remark The ab ove is stated for Lipschitz continuous maps to simplify the state

ment For a sp ecic and a sp ecic the following condition instead of the Lipschitz

condition ab ove is sucient for the existence of F

df x f x k whenever dx x k

for a certain nite set of integers k more precisely for k

and similarly for the isomorphism in the second part When X is compact and is

f implies that this condition is satised for suciently small given the continuityof

s Use the continuity of the distance function d X X R and the compactness

half of the prop osition there of the diagonal set X X And in the second

are cases when the equality F G holds without comp osing with the relaxcontrol

map eg see

A RANICKI AND M YAMASAKI

Weareinterested in the limit of controlled Lgroups

Denition Let p M X b e a control map

Let b e p ositivenumb ers such that We dene

L X p imfL X p L X p g

X n X n X

For we dene the stable controlled Lgroup of X with co ecient p by

L X p L X p

X X

n n

The controlled Lgroup with co ecient p is dened by

L X p L X p lim

X X

n n

where the limit is taken with resp ect to the obvious relaxcontrol maps

L X p L X p

X X

n n

In section we study a certain stability result for the controlled Lgroups in

some sp ecial case

Epsiloncontrolled pro jective Lgroups

Fix a subset Y of X and let F be a family of subsets of X such that Z Y

for each Z F In this section we intro duce intermediate controlled Lgroups

L Y p which will app ear in the stableexact sequence of a pair and also in

the MayerVietoris sequence Roughly sp eaking these are dened using controlled

pro jective quadratic chain complexes C pwithvanishing controlled reduced

for each pro jective class C p K Z p n Ranicki and Yamasaki

Z F Here p denotes the restriction p jp Z p Z Z of p as in the

Z X X

X X

previous section

For a pro jectivemoduleA p on p its dual A p is the pro jectivemodule

A p on p If f A p B q is an morphism then f B q

A p is also an morphism For an pro jectivechain complex on p

C p C p C p

r r r r

in the sense of C p will denote the pro jectivechain complex on p dened

CONTROLLED LTHEORY

nr nr

C p C p

nr nr

An ndimensional quadratic structure on a pro jectivechain complex C pon

p is an ndimensional quadratic structure on C in the sense of x suchthat

nr s

and r Z Similarly C p C pisan morphism for every s

s r

an n dimensional quadratic structure on a chain map f C p D q

is an n dimensional quadratic structure on f C D such that

nr s nr s

D q D q and C p C pare morphisms for

s r s r

every s and r Z An ndimensional pro jectivechain complex C p on p

equipp ed with an ndimensional quadratic structure is called an ndimensional

pro jective quadratic complex on p andan chain map f C p D q between

an ndimensional pro jectivechain complex C ponp and an n dimensional

pro jective chain complex D q on p equipp ed with an n dimensional

quadratic structure is called an n dimensional pro jective quadratic pair on p

An cob ordism of ndimensional pro jective quadratic complexes C p

C p onp is an n dimensional pro jective quadratic pair on p

X X

f f C p C p D q

with b oundary C p C p

An ndimensional quadratic structure on C pis Poincare if

C pC T C p C p

contractible C p is Poincare if is Poincare Similarly an n is

dimensional quadratic structure onf C p D q is Poincare if C p

and

D q C T f T C f D q

e are b oth contractible A pair f is Poincare if is Poincar

Let Y and b e a subset of X and F beafamilyofsubsetsofX such that Z Y

for every Z F

Denition Let n and L Y p is the equivalence classes of n

dimensional Poincare pro jective quadratic complexes C ponp suchthat

C p in K Z p n for each Z F The equivalence relation is generated by

Poincare cob ordisms f f C p C p D q on p

such that D q in K Z p n for each Z F When F fX gwe omit

fX g

the braces and write L Y p instead of L Y p When F fg then we

X X

use the notation L Y p since it dep ends only on p

Y Y n

A RANICKI AND M YAMASAKI

Prop osition Direct sum induces an ab elian group structure on L Y p

Furthermore if

C pC p L Y p

then there is a Poincare cob ordism on p

f f C p C p D q

for each Z F suchthat D q in K Z p n

A functoriality with resp ect to maps and relaxation of control similar to holds

for epsilon controlled pro jective Lgroups

Prop osition Let F f f b e a map from p M X to p N Y and

X Y

supp ose that f is Lipschitz continuous with Lipschitz constant ie there exists a

constant suchthat

df x f x dx x x x X

If f A B and there exists a Z F satisfying f Z Z for each Z F

then F induces a homomorphism

F F

B p F L A p L

Y X

n n

Remark As in the remark to for a sp ecic and a we do not need the full

Lipschitz condition to guarantee the existence of F

There is an obvious homomorphism

Y p C C L Y p L

X n Y

On the other hand the controlled K theoretic condition p osed in the denition can

b e used to construct a homomorphism from a pro jective Lgroup to a free Lgroup

Prop osition There exist a constant such that the following holds true

for anycontrol map p M X any subset Y X any family of subsets F of X

containing Y any element Z of F anynumber n and any p ositivenumbers

such that there is a welldened homomorphism functorial with resp ect to

relaxation of control

i L Y p L Z p

Z X n Z

such that the following comp ositions are equal to the maps induced from inclusion

maps

F fZ g

Y p Z p L L Z p L

X Z n Z

n n

L Y p L Y p L Z p

X n Y n Z

Remark Actually works

Stablyexact sequences

CONTROLLED LTHEORY

In this section we describ e two stablyexact sequences The rst is the stablyexact

sequence of a pair

i i

X X

L X p L Y p L X Y p L Y p

n X X n X n X

n n

where the dotted arrows are only stably dened The precise meaning will b e ex

plained b elow The second is the MayerVietoristyp e stablyexact sequence

L A p L B p L C p L X p

n A n B X n X

L C p

where X A B C A B and F fA B g

Fix an integer n let Y Z be subsets of X and let be three

n n n n n

p ositivenumb ers satisfying

n n n n

where is the number p osited in Then there is a sequence

i i

Y p L X p L X Z p L

n X n n X n n n X n

where i is the homomorphism given in and j is the homomorphism induced by

the inclusion map and relaxation of control The subscripts are there just to remind

the reader of the degrees of the relevant Lgroups

Theorem There exist constants which do not dep end on

p suchthat

n n

if n Z Y and then the following comp osition j i is

n n n n

zero

j i L Y p L X p L X Z p

n X n n X n n n X n

n n

if n Y Z and then there is a connecting homomor

n n n n

phism

L Y p L X Z p

n n X n n n X

such that the following comp osition j is zero

Y p L X Z p L j L X p

n X n n n X n n X n

and if so that the homomorphism i is welldened the following

n n

comp osition i is zero

Y p i L X Z p L L X p

n X n n n X n n X n

A RANICKI AND M YAMASAKI

Theorem There exist constants which do not dep end on

p suchthat

n n

if n so that i is welldened Z Y

n n n n

n n n

the and so that is welldened then Y Z

n n n n

image of the kernel of i in L Y p is in the image of

n n n

L X p

L Y p

n X n

L X Z p

L Y p

n X

n n

n n n

n n

if n so that j is welldened Y Z and

n n n n

n n n

so that i is welldened then the image of the kernel of j in

n n

L X p is in the image of i

n X

L X p L X Z p

n X n n n X n

L X p

L Y p

n X

n n n

if n so that is welldened and Z

n n n n n

n n

Y then the image of the kernel of in L X Z p is in the image

n X

n n

of j

L X Z p L Y p

n n X n n X n

L X Z p L X p

n X n X

n n n

Here the vertical maps are the homomorphisms induced by inclusion maps and relax

ation of control

Next weinvestigate the MayerVietoristyp e stablyexact sequence Fix an inte

ger n and assume that X is the union of two closed subsets A and B with

n n

intersection C A B Supp ose three p ositivenumb ers satisfy

n n n n n n

n n n n

and dene a family F to b e fA B g Then wehave a sequence

n n n

L C p L A p L B p L X p

n X n n n A n n n B n n X n

n n n

CONTROLLED LTHEORY

Theorem There exist constants which do not dep end on

p suchthat

if n and then the following comp osition j i is zero

n n n

L X p C p L A p L B p L

n X n n X n n n A n n n B n

n n n

n n

if n C C and if we set

n n n n

F fA A C B B C g

n n n n n n n

then there is a connecting homomorphism

C p L X p L

n X n n X n

is zero such that the following comp osition j

L X p L C p L A p L B p

n X n n X n n n n n n n

and if so that the homomorphism i is welldened the following

n n

comp osition i is zero

C L X p L p

n n X n X n

L A p L B p

n n n n n n

Theorem There exist constants which do not dep end on

p suchthat

n n

if n so that i is welldened C C

n n n n

n n n

so that is welldened then the image of the kernel of i in

n n

C p is in the image of L

n n

L A p L B p

L C p

n n n n n n

n X n

L X p

n X

C p L

n n

if n so that j is welldened C C

n n n n

n n n n n

so that i is welldened and F fA A C B B C gthenthe

n n

n n n n n

p L B p is in the image of i image of the kernel of j in L A

n n

n n n n

L A p L B p L X p

n n n n n n n X n

L A p L B p

n n

L C p

n n n n

n n

A RANICKI AND M YAMASAKI

n n

if n C C so that is welldened

n n n n n n

n n

C C A A C and B B C thentheimageofthekernel

n n

n n n n n

of in L X p is in the image of j

n X

L X p

C p L

n X n

L X p L A p L B p

n X n n

n n n n n

Here the vertical maps are the homomorphisms induced by inclusion maps and relax

ation of control

Theorems are all straightforward to prove

Lo callynite analogues

Up to this p oint we considered only nitely generated mo dules and chain complexes

To study the behaviour of controlled Lgroups we need to use innitely generated

ob jects such ob jects arise naturally when wetake the pullback of a nitely generated

ob ject via an innitesheeted covering map

Consider a control map p M X and take the pro duct with another metric

space N

p M N X N

X N

Here we use the maximum metric on the pro duct X N

Denition Ranicki and Yamasaki p A geometric mo dule on the pro duct

space M N is said to be M nite if for any y N there is a neighbourhood U

of y in N suchthat M U contains only nitely many basis elements a pro jective

mo dule A p on M N is said to be M nite if A is M nite a pro jectivechain

complex C p on M N is M nite if each C p is M nite In we used

r r

the terminology M lo cally nite but this do es not sound right and we decided to

use M nite instead N lo cally M nitemay b e describing the meaning b etter

butitistoolong When M is compact M niteness is equivalent to the ordinary

lo callyniteness

Denition Using this notion one can dene M nite controlled Lgroups L X

N Y N p and M nite controlled pro jective Lgroups L Y N p

X N X

by requiring that every chain complexes concerned are M nite

Consider the case when N R We would like to apply the M nite version

of the MayerVietoristyp e stable exact sequence with resp ect to the splitting R

The following says that one of the three terms in the sequence

vanishes

CONTROLLED LTHEORY

Prop osition Let p M X be a control map For any andr R

M M

L X rp L X r p

X X

n n

M M

e e

K X rp nK X r p n

X X

Pro of This is done using rep eated shifts towards innity and the Eilenb erg Swindle

Let us consider the case of L X r p Let J r and dene

T M J M J by T x tx t Take an elementc L X Jp

It is zero b ecause there exist M nite Poincare cob ordisms

c T c c c T c T c T

c T c T c T c

Thus the MayerVietoris stablyexact sequence reduces to

L X Rp L X I p

X R X I

where I for some A diagram chase shows that there exists

awelldened homomorphism

L X I p L X Rp

X I X R

where The homomorphisms and are stable inverses of each

n n n

other the comp ositions

M M

L X Rp L X Rp

X R X R

n n

p p

X I p X I p L L

X I n X I

n n

trol maps are b oth relaxcon

Note that for any a pro jective Lgroup analogue of gives an isomorphism

p p

X fgp X I p L L

X X I

n n

In this case no comp osition with relaxcontrol map is necessary b ecause X I is

given the maximum metric Thus wehave obtained

Theorem There is a stable isomorphism

X Rp L X p L

X R X

Similarlywehav e

A RANICKI AND M YAMASAKI

Theorem There is a stable isomorphism

plf

L X Rp L X p

X R X

Stability in a sp ecial case

In this section we treat the sp ecial case when the control map is the identity map

The following can b e used to replace the controlled pro jective Lgroup terms in the

previous section by controlled Lgroups

Prop osition Supp ose that Y X is a compact p olyhedron or a compact

metric ANR emb edded in the Hilb ert cub e and that p is the identity map on Y

Y Y

Then for any and n there exists a suchthatfor any p ositivenumber

satisfying there is a welldened homomorphism functorial with resp ect to

relaxation of control

L Y p L Y

X n Y

such that the comp ositions

F F

Y p Y p L L Y L

X X n Y

n n

Y p L L Y L Y

X n Y n Y

are both relaxcontrol maps In particular L Y and L Y are stably

Y n Y

isomorphic

Pro of Let where is the p ositivenumb er p osited in By and

of there exists a such that the following map is a zero map

e e

K Y n K Y n C p C p

Y Y

Therefore if there is a homomorphism

F FfY g

Y p L Y p C p C p L

X X

n n

The desired map is obtained by comp osing this with the map

FfY g

i L Y p L Y

Y X n X

corresp onding to the subspace Y

Remark If Y is a compact p olyhedron then there is a constant which

dep ends on n and Y such that ab ove can b e taken to b e For this we need

to change the statement and the pro of of of like those of b elow

CONTROLLED LTHEORY

Recall that in our MayerVietoristyp e stablyexact sequence each piece of space

tends to get bigger in the pro cess The following can b e used to remedy this in certain

cases It is stated here for the identitycontrol map but there is an obvious extension

to general control maps

Prop osition Let r X A b e a strong deformation retraction with aLips

chitz continuous strong deformation of Lipschitz constant and i A X b e the

inclusion map Then r and i induce stable isomorphisms of controlled Lgroups in

the following sense if then for any the comp ositions

r i

L X L A L X

n X n A n X

i r

L A L X L A

n A n X n A

are relaxcontrol maps

Pro of Obvious from

Theorem Fix a compact p olyhedron X andaninteger n Then there exist

numbers and which dep end on n X and the triangulation

such that for any subp olyhedrons A and B of X anyinteger k and anynumber

there exists a ladder

lf k lf k lf k lf k

L C R L A R L B R L K R

n n n n

lf k lf k lf k lf k

L C R L A R L B R L K R

n n n n

i j

lf k lf k lf k

L C R L A R L B R

n n n

lf k lf k lf k

L C R L A R L B R

n n n

which is stablyexact in the sense that

the image of a horizontal map is contained in the kernel of the next map and

the relaxcontrol image in the second rowofthekernel of a map in the rst row

is contained in the image of a horizontal map from the left

where C A B and K A B and the vertical maps are relaxcontrol maps

Pro of This is obtained from the lo callynite versions of combined with

and the strong deformations of the neighbourhoods of A and B in K

only nitely many can be chosen to be PL and hence Lipschitz Since there are

subp olyhedrons of X with a xed triangulation wemaycho ose constants and

indep endentofA and B

A RANICKI AND M YAMASAKI

Theorem Supp ose X is a compact p olyhedron and n is an integer Then

there exist numb ers and which dep end on X and n suchthat

L X L X

X X

n n

for every and every

Pro of We inductively construct sequences of p ositivenumbers

such that for any sub complex K of X with the numb er of simplices l

if andk then

l l

lf k lf k

L K R L K R

K K l

n n

and

if then the homomorphism

lf k lf k

L K R L K R

K K

n n

is injective

Here R is given the maximum metric

First supp ose l ie K is a single p oint Any ob ject with b ounded control

on R can b e squeezed to obtain an arbitrarily small control therefore

the numb er p osited in

works

Next assume wehave constructed and for i l We claim that

i i

min f g

l l l

satisfy the required condition Supp ose the numb er of simplices of K is less than or

equal to l Cho ose a simplex of K of the highest dimension and call the simplex

viewed as a subp olyhedron A and let B K intA Supp ose and

A diagram chase starting from an elementof

k lf

K R L

in the following diagram establishes the prop erty Here the entries in eachofthe

columns are

lf k lf k lf k

L A R L B R L C R

n n n

k k lf k lf lf

B R A R L K R and L L

n n n

CONTROLLED LTHEORY

for various s sp ecied in the diagram

lf lf lf lf lf lf

L A L B L K L C L A L B

n n n n n n

l l

l l l

Next supp ose A diagram chase starting from an elementof

k lf

K R L

representing an elementof

k k lf lf

K R g K R L kerfL

n n

establishes

lf lf lf lf lf

L C L A L B L K L C

n n n n n

l l

l l l

l l

A RANICKI AND M YAMASAKI

References

Quinn F Resolutions of homology manifolds and the top ological characteriza

tion of manifolds Invent Math

Ranicki A AExact Sequences in the Algebraic Theory of Surgery Math Notes

vol Princeton Univ Press

and Yamasaki M Symmetric and quadratic complexes with geometric con

trol Pro c of TGRCKOSEF vol available electronically on

WWW from httpwwwmathsedacukaar and also from httpmath

josaiacjpyamasaki

and Controlled K theoryTop ology Appl

Yamasaki M Lgroups of crystallographic groups Invent Math

Andrew Ranicki

Department of Mathematics and Statistics

UniversityofEdinburgh

Edinburgh EH JZ

Scotland

email aarmathsedacuk

Masayuki Yamasaki

Department of Mathematics

Josai Universit y

Keyakidai Sakado

Saitama

Japan

email yamasakimathjosaiacjp

NOTES ON SURGERY AND C ALGEBRAS

JOHN ROE

Introduction

A C algebra is a complex BanachalgebraA with an involution which satises

the identity

kx xk kxk x A

The study of C algebras seems to b elong entirely within the realm of functional

analysis but in the past twentyyears they haveplayed an increasing role in geo

metric top ology The reason for this is that C algebra K theory is a natural

receptacle for higher indices of elliptic op erators including the higher signatures

which feature as surgery obstructions The big picture was originated byAtiyah

and Connes in these notes based on my talk at the Josai conference I

want to explain part of the connection with particular reference to surgery theory

For more details one could consult

About C algebras

The following are key examples of C algebras

The algebra C X of continuous complexvalued functions on a compact Haus

dor space X

The algebra BH of b ounded linear op erators on a Hilb ert space H

Gelfand and Naimark ab out proved Anycommutative C algebra with

unit is of the form C X any C algebra is a subalgebra of some BH

Let A b e a unital C algebra Let x A be normalthatisxx x x Then x

generates a commutative C subalgebra of A whichmustbeoftheformC X In

fact we can identify X as the spectrum

xf C x has no inverseg X

with x itself corresp onding to the canonical X C

Hence we get the Spectral Theorem for any C x we can dene x A

so that the assignment x is a ring homomorphism

If x is self adjoint x x then x R

One can dene K theory groups for C algebras For A unital

K A Grothendieck group of fg pro jective Amo dules

K A GL A

with a simple mo dication for nonunital A These groups agree with the ordinary

top ological K theory groups of the space X in case A is the commutative C algebra

C X

For anyinteger i dene K K Then to any short exact sequence of C

i i

algebras

The hospitality of Josai University is gratefully acknowledged

JOHN ROE

J A AJ

there is a long exact K theory sequence

K J K A K AJ K J

i i i i

The p erio dicityisa version of the Bott p erio dicity theorem Notice that algebraic

K theory do es not satisfy Bott p erio dicity analysis is essential here

A go o d reference for this material is

Classical Fredholm theory provides a useful example of C algebra K theory at

work Recall that an op erator T on a Hilb ert space H is called Fredholm if it has

nitedimensional kernel and cokernel Then the index of T is the dierence of the

dimensions of the kernel and cokernel

Definition The algebra of compact op erators KH is the C algebra

generatedbytheoperators with nitedimensional range

Compact and Fredholm op erators are related by Atkinsons Theorem which

states that T BH is Fredholm if and only if its image in BH KH is

invertible

Thus a Fredholm op erator T denes a class T in K BK Under the connect

ing map this passes to T K KZ this is the index

Abstract signatures

Recall that in symmetric Ltheory wehave isomorphisms L Z L R Z

The second map asso ciates to a nonsingular real symmetric matrix its signature

Numb er of p ositive eigenvalues Numb er of negativeeigenvalues

Can we generalize this to other rings

If M is a nonsingular symmetric matrix over a C algebra A we can use the

sp ectral theorem to dene pro jections p and p corresp onding to the p ositiveand

negative parts of the sp ectrum Their dierence is a class in K A

This pro cedure denes a map L A K A for every C algebra A and it

canbeshown that this map is an isomorphism There is a similar isomorphism

on the level of L

Now let b e a discrete group The group ring Z acts faithfully by convolution

on the Hilb ert space The C subalgebra of B generated by Z acting in

this way is called the group C algebra C

Wehave a map L Z K C

Gelfand and Mishchenko observed that this map is a rational isomorphism

for free ab elian Then C C T byFourier analysis

Remark Our map from L to K is sp ecial to C algebras if it extended naturally

to a map on all rings wewould have for a free ab elian group a diagram

L C L Z

K Z K C

NOTES ON SURGERY AND C ALGEBRAS

Going round the diagram via the top rightwe get Gelfand and Mishchenkos map a

rational isomorphism But the b ottom lefthand group is of rank one by the Bass

HellerSwan theorem Chapter XI I This contradiction shows that the lefthand

vertical map cannot exist

The signature operator

Let M be a complete oriented Riemannian manifold of even dimension for

on L dierential forms where simplicity Dene the op erator F D D

D d d d exterior derivative d its adjoint

F is graded byaninvolution i here i andthepower dep ends

on the dimension and the degree of forms see for the correct formula Thus

graded it is called the signatureoperator

If M is compact then F is Fredholm Moreover the index of F is the signature

of M This is a simple consequence of Ho dge theory

Remark The choice of normalizing function xx x in F D

do es not matter as long as it has the right asymptotic b ehaviour

Consider now the signature op erator on the universal cover M of a compact

manifold M F b elongs to the algebra A of M equivariant op erators More

over it is invertible mo dulo the ideal J of equivariant local ly compact op erators

This follows from the theory of elliptic op erators

Thus via the connecting map K AJ K J we get an index in K J

C Lemma J C K Consequently K J K

r r

We have dened the analytic signature of M as an element of K C In

general it can b e dened in K M wherei is the dimension of M mo d

Proposition The analytic signature is the image of the Mishchenko

Ranicki symmetric signature under the map L K

Corollary The analytic signature is invariant under orientation preserv

ing homotopy equivalence

Direct pro ofs of this can b e given

We can now dene an analytic surgery obstruction dierence of analytic

signatures for a degree one normal map

Can we mimic the rest of the surgery exact sequence

K homology

Let A be a C algebra A Fredholm module for A is made up of the following

things

A representation A BH ofA on a Hilb ert space

An op erator F BH such that for all a A the op erators

Fa aF F a F F a

b elong to KH

The signature op erator is an example with A C M

One can dene b oth graded and ungraded Fredholm mo dules These ob jects

can b e organized into Grothendieck groups to obtain Kasparovs K homology groups

K A i for graded and i for ungraded mo dules They are

contravariant functors of A

JOHN ROE

Remark The critical condition in the denition is that F a K for all a One

should regard this as a continuous control condition In fact if A is commutativeit

was shown by Kasparov that the condition is equivalenttof Fg K whenever

f and g have disjoint supp orts which is to say that F has only nite rank

propagation b etween op en sets with disjoint closures

Kasparov proved that the name K homology is justied

Theorem Let A C X be a commutative C algebra Then

K A is natural ly isomorphic to H X K C the topological K homology of X

We assume that X is metrizable here If X is a bad space not a nite complex

then H refers to the Steenro d extension of K homology if X is only local ly

compact and we take A C X the continuous functions vanishing at innity

then we get lo cally nite K homology

Kasparovs denition was reformulated in the language of duality byPaschke

and Higson For a C algebra A and ideal J dene the algebra AJ to

consist of those T BH suchthat

T a K a A and

Tj K j J

where is a go o d ie suciently large representation of A on H

Proposition Paschke duality theorem There is an isomorphism

K AK AAA

for al l separable C algebras A

Let us intro duce some notation For a lo cally compact space X write X

for C X we call this the algebra of pseudolocal op erators and X

for C X C X the algebra of lo cally compact op erators

Nowlet X M the universal cover of a compact manifold M as ab ove and

consider the exact sequence

f f

M M M M

The sup erscript denotes the equivariant part of the algebra Wehave incor

p orated into the sequence the fundamental isomorphism

f f

M M M M

which exists b ecause b oth sides consist of local ob jects formal symb ols in some

sense and there is no diculty in lifting a lo cal ob ject from a manifold to its

universal cover

Note that M lo cally compact op erators Thus applying the K theory

functor we get a b oundary map

A K M K M M K C

i i

This analytic assembly map takes the homology class of the signature op erator F to the analytic signature

NOTES ON SURGERY AND C ALGEBRAS

On the Novikov conjecture

We use the ab ove machinery to make a standard reduction of the Novikov

conjecture Assume B is compact and let f M B Consider the diagram

H M Q K M K C

f f

H B Q K B

By the AtiyahSinger index theorem f chF is Novikovs higher signature the

push forward of the Poincare dual of the Lclass So if A K B K C

is injective the Novikov conjecture is true for

This has led to a number of partial solutions to the Novikov conjecture using

analysis Metho ds used have included

Cyclic cohomology pair K C with H B R Need suitable

dense subalgebras very delicate

Kasparov KKtheory sometimes allows one to construct an inverse

of the assembly map as an analytic generalized transfer

Control led C algebratheory parallel developmenttocontrolled top ol

ogy see later

The analytic structure set

Recall the exact sequence

K C K M K M K C

i i i

r r

The analogy with the surgery exact sequence suggests that we should think of

K M as the analytic structure set of M

Example Supp ose M is spin Then one has the Dirac op erator D and one can

normalize as b efore to get a homology class

F F D

If M has a metric of p ositive scalar curvature then byLichnerowicz there is a

gap in the sp ectrum of D near zero Thus we can cho ose the normalizing function

so that F exactly Then F K M gives the structure invariant of

the p ositive scalar curvature metric

Notice that Lichnerowicz vanishing theorem now follows from exactness in

the analytic surgery sequence

It is harder to give a map from the usual structure set to the analytic one In

the same way that the p ositive scalar curvature invariantgives a reason for the

for Lichnerowicz vanishing theorem we want an invariant which gives a reason

the homotopyinvariance of the symmetric signature

Here is one p ossibility Recall Pedersens description in these pro ceedings of

TOP

the structure set S M as the Ltheory of the category

f f

B M I M Z

Replacing Z by C wehave a category

whose ob jects can b e completed to Hilb ert spaces with C M action

whose morphisms are pseudolo cal

JOHN ROE

Using Voiculescus theorem which says that all the ob jects can be emb edded

more or less canonically in a single suciently large representation of C M

we should get a map from the structure set to K M However there is

a signicant problem Are the morphisms b ounded op erators Similar questions

seem to come up elsewhere if one tries to use analysis to study homeomorphisms

and one needs some kind of torus trick to resolve them compare

Controlled C algebras

DI F F

A more direct approach can b e given to obtaining a map from S M

Let W b e a metric space noncompact and supp ose C W BH as usual

An op erator T on H is boundedly control led if there is R R T such that

T whenever distance from Supp ort to Supp ort is greater than R

Example If D is a Diractyp e op erator on complete Riemannian M and has

compactly supp orted Fourier transform then D is b oundedly controlled

Dene W j to be the C subalgebras generated by b oundedly

controlled elements Then from the ab ove one has that a Dirac typ e op erator

on a complete Riemannian manifold W has a b oundedly controlled index in

K W

In fact al l elliptic op erators have b oundedly controlled indices in full generality

one has a b ounded assembly map A K W K W and

the assembly of the signature op erator is the bounded analytic signature

This b ounded analytic signature can also be dened for suitable b ounded

b ounded Poincare complexes b ounded in b oth the analytic and geometric senses

If W has a compactication X W Y whichissmallatinnity then there

is a close relation b etween b ounded and continuously controlled C algebra theory

In fact consider a metrizable pair X Y let W X n Y We can dene

continuously controlled C algebras W Then one has

Proposition We have

W X C X

W C X C W

The result for W is an analytic counterpart to the theorem control means

homology at innity compare

in Now we can dene our map from the structure set for simplicity we work

the simply connected case Given a homotopy equivalence f M M formthe

double trump et space W consisting of op en cones on M and M joined by the

mapping cylinder of f there is a picture in This is a b ounded b ounded

Poincare space with a map to M R continuously controlled by M S

Thus we have the analytic signature in K X R Map this by the

comp osite

X R X I X X R

cc cc

using the preceding prop osition The image is the desired structure invariant

The various maps wehave dened t into a diagram relating the geometric and

C surgery exact sequences The diagram commutes up to some factors of

arising from the dierence b etween the Dirac and signature op erators

NOTES ON SURGERY AND C ALGEBRAS

Final remarks

C surgery can pro duce some information in a wide range of problems

SurjectivityofC assembly maps is related to representation theory

Some techniques for Novikov are only available in the C world

But We dont understand well how to do analysis on top ological manifolds

Top ologists construct analysts only obstruct

References

MF Atiyah Global theory of elliptic op erators In Proceedings of the International Sympo

sium on Functional Analysis pages UniversityofTokyo Press

MF Atiyah Elliptic op erators discrete groups and von Neumann algebras Asterisque

MF Atiyah and IM Singer The index of elliptic op erators I I I Annals of Mathematics

H Bass Algebraic K Theory Benjamin

A Connes A survey of foliations and op erator algebras In Operator Algebras and Appli

cations pages American Mathematical So ciety Pro ceedings of Symp osia in

Pure Mathematics

A Connes NonCommutative Geometry Academic Press

A Connes M Gromov and H Moscovici Group cohomology with Lipschitz control and

higher signatures Geometric and Functional Analysis

A Connes and H Moscovici Cyclic cohomology the Novikov conjecture and hyp erb olic

groups Topology

SC Ferry Remarks on Steenro d homologyInSFerryARanicki and J Rosenb erg editors

Proceedings of the Oberwolfach Conference on the Novikov Conjecturevolume of

LMS Lecture Notes pages

IM Gelfand and AS Mishchenko Quadratic forms over commutative group rings and K

theory Functional Analysis and its Applications

algebras and controlled top ology K Theory N Higson EK Pedersen and J Ro e C

To app ear

N Higson and J Ro e Mapping surgery to analysis In preparation

J Kaminker and JG Miller Homotopyinvariance of the index of signature op erators over

C algebras Journal of Operator Theory

J Kaminker and C Scho chet K theory and Steenro d homology applications to the Brown

DouglasFillmore theory of op erator algebras Transactions of the American Mathematical

Society

GG Kasparov Top ological invariants of elliptic op erators I K homology Mathematics of

the USSR Izvestija

GG Kasparov The op erator K functor and extensions of C algebras Mathematics of the

USSR Izvestija

GG Kasparov Equivariant KKtheory and the Novikov conjecture Inventiones Mathemat

icae

A Lichnerowicz Spineurs harmoniques Comptes Rendus de lAcademie des Sciences de

Paris

C Ogle Cyclic homology and the strong Novikov conjecture Preprint

W Paschke K theory for commutants in the Calkin algebra Pacic Journal of Mathematics

EK Pedersen J Ro e and S Weinb erger On the homotopyinvariance of the b oundedly

controlled analytic signature of a manifold over an op en cone In S Ferry A Ranicki

andJRosenb erg editors Proceedings of the Oberwolfach Conference on the Novikov

Conjecturevolume of LMS Lecture Notes pages

EK Pedersen and CA Weib el A nonconnective delo oping of algebraic K theory In

volume AA Ranicki et al editor Algebraic and Geometric Topology Rutgers

of Springer Lecture Notes in Mathematics pages

J Ro e Coarse cohomology and index theory on complete Riemannian manifolds Memoirs

of the American Mathematical Society

JOHN ROE

J Ro e Index theory coarse geometry and the topology of manifoldsvolume of CBMS

ConferenceProceedings American Mathematical So ciety

J Rosenb erg Analytic Novikov for top ologists In S Ferry A Ranicki and J Rosenberg

editors Proceedings of the Oberwolfach Conference on the Novikov Conjecturevolume

of LMS Lecture Notes pages

NE WeggeOlsen K theory and C algebras Oxford University Press

Jesus College Oxford OX DW England

Email address jroejesusoxacuk

CHARACTERIZATIONS OF INFINITEDIMENSIONAL

MANIFOLD TUPLES AND THEIR APPLICATIONS TO

HOMEOMORPHISM GROUPS

TATSUHIKOYAGASAKI

Chapter Characterization of s S S manifolds

The purp ose of this article is to survey tuples of innitedimensional top ological

manifolds and their application to the homeomorphism groups of manifolds

s S S manifolds

A top ological E manifold is a space which is lo cally homeomorphic to a space E

In this article all spaces are assumed to b e separable and metrizable In innite

dimensional top ological manifold theorywe are mainly concerned with the following

mo del spaces E

i the compact mo del the Hilb ert cub e Q

ii the complete linear mo del the Hilb ert space

The Hilb ert space or more generally any separable Frechet space is homeo

morphic to s If we regard s as a linear space of sequences of

real n umb ers then it contains several natural incomplete linear subspaces

iii the big sigma fx s sup jx j g the subspace of bounded

n n

sequences

iv the small sigma fx s x for almost all ng the subspace of

n n

nite sequences

The main sources of innitedimensional manifolds are various spaces of func

tions emb eddings and homeomorphisms In Chapter we shall consider the group

of homeomorphisms of a manifold When M is a PLmanifold the homeomorphism

group H M contains the subgroup H M consisting of PLhomeomorphisms of

the ambient M and we can ask the natural question How is H M sited in

group H M This sort of question leads to the following general denition An

l tuple of spaces means a tuple X X X consisting of an ambientspace

X and l subspaces X X

Mathematics Subject Classication N N

Key words and phrases Innitedimensional manifolds Strong universality Homeomorphism

groups Top ological manifolds

TATSUHIKOYAGASAKI

Denition A tuple X X X is said to be an E E E manifold if

l l

for every p oint x X there exist an op en neighborhood U of x in X and an op en

set V of E such that U U X U X V V E V E

l l

In this article we shall consider the general mo del tuple of the form s S S

where S S are linear subspaces of s Some typical examples are

v the pairs s s

vi the triples s s s s and s

f f

where a s are the countable pro duct of s and resp ectively

b fx x for almost all ng fx x for

n n n n

f f

almost all ng

Note that s s s s ands

f f f f

s The statements and follow from the characterizations of

manifolds mo deled on these triples x Theorems In Section

we shall give a general characterization of s S S manifold under some nat

ural conditions on the mo del s S S

Basic properties of infinitedimensional manifolds

In this section we will list up some fundamental prop erties of innitedimensional

manifolds We refer to for general references in innitedimensional

manifold theory

Stability

Since s is a countable pro duct of the interval it is directly seen that

X for every s s Applying this argument lo cally it follows that X s s

manifold X cf More generallyit has b een shown that if X X X is an

X X X This prop erty s manifold then X s X X

is one of characteristic prop erties of innitedimensional manifolds To simplify the

notation we shall use the following terminology

Denition WesaythatX X X isE E E stable if X E X

l l

X X X E X E

l l l

Homotopy negligibility

Denition A subset B of Y is said to b e homotopy negligible hn in Y if there

exists a homotopy Y Y such that id and Y Y n B t

t t

In this case wesaythat Y n B has the homotopy negligible hn complementin

When Y is an ANR B is homotopy negligible in Y i for every op en set U

of Y the inclusion U n B U is a weak homotopy equivalence Again using the

innite co ordinates of s we can easily verify that has the hn complement in

INFINITEDIMENSIONAL MANIFOLD TUPLES

s Therefore it follows that if X X isans manifold then X has the hn

complementinX

General p osition prop erty Strong universality

Z embedding approximation in smanifolds

The most basic notion in innitedimensional manifolds is the notion of Z sets

Denition A closed set Z of X is said to be a Z set a strong Z set of X if

for everyopencover U of X there is a map f X X such that f X Z

cl f X Z and f id U

Here for an op en cover V of Y twomap f g X Y are said to b e V close

and written as f g V if for every x X there exists a V V with f x

g x V Using the innite co ordinates of s wecan showthe following general

p osition prop ertyof smanifolds

Facts Supp ose Y is an smanifold Then for every map f X Y from a

separable completely metrizable space X and for every op en cover U of Y there

exists a Z emb edding g X Y with f g U Furthermore if K is a closed

subset of X and f j K Y is a Z emb edding then we can take g so that

g j f j

K K

Strong universality

To treat various incomplete submanifolds of smanifolds manifolds mani

folds etc we need to restrict the class of domain X in the ab ove statement Let

C b e a class of spaces

Denition M Bestvina J Mogilski et al

AspaceY is said to b e strongly C universal if for every X Cevery closed subset

K of X every map f X Y such that f j K Y is a Z emb edding and

for every op en cover U of Y there exists a Z emb edding g X Y such that

g j f j and f g U

K K

In some cases the ab oveemb edding approximation conditions can b e replaced

by the following disjointapproximation conditions

Denition Wesay that a space X has the strong discrete approximation prop erty

or the disjoint discrete cells prop erty if for every map f Q X of a

i i

countable disjoint union of Hilb ert cub es into X and for every op en cover U of X

g Q X suchthatf g U and fg Q g is discrete there exists a map

i i i i

in X

TATSUHIKOYAGASAKI

Strong universality of tuples R Cauty J BaarsH GladdinesJ van

Mill et al

A map of tuples f X X X Y Y Y issaidtobelayer preserv

l l

ing if f X n X Y n Y for every i l where X X X

i i i i l

Let M b e a class of l tuples of spaces

ersal if Denition An l tuple Y Y Y is said to b e strongly Muniv

it satises the following condition

for every tuple X X X M every closed subset K of X every

map f X Y such that f j K K X K X Y Y Y is a

K l l

layer preserving Z emb edding and every op en cover U of Y there exists a layer

preserving Z emb edding g X X X Y Y Y suchthat g j f j

l l K K

and f g U

In Section we shall see that the stabilityhn complement implies the

strong universality

Uniqueness prop erties of absorbing sets

The notion of hn complement can be regarded as a homotopical absorbing

prop erty of a subspace in an ambient space The notion of strong universalityof

tuples also can be regarded as a sort of absorption prop erty combined with the

general p osition prop erty Roughly sp eaking for a class ManMabsorbing set of

an smanifold X is a subspace A of X suchthati A has an absorption prop ertyin

X for the class Mii A has a general p osition prop ertyfor M and iii A b elongs

to the class M The notion of strong universality of tuples realizes the conditions

i and ii simultaneously The condition iii usually app ears in the form A is a

countable union of Z sets of A whichbelongtoM The most imp ortant prop erty

of absorbing sets is the uniqueness prop erty This prop erty will playakey role in

the characterizations of tuples of innitedimensional manifolds

Capsets and fd capsets RD Anderson and TA Chapman

The most basic absorbing sets are capsets and fd capsets A space is said to b e

compact fdcompact if it is a countable union of compact nitedimensional

compact subsets

Denition Supp ose X is a Qmanifold or an smanifold A subset A of X is said

to b e a fd capset of X if A is a union of fd compact Z sets A n which

satisfy the following condition for every every fd compact subset K of X

K A suchthat and every n there exist an m n and an emb edding h

i dh id and ii h id on A K

K n

For example is a capset of s and is fd capset of s The fd capsets havethe

following uniqueness prop erty

INFINITEDIMENSIONAL MANIFOLD TUPLES

Theorem If A and B are fd capsets of X then for every open cover U of

X there exists a homeomorphism f X A X B with f id U

Absorbing sets in smanifolds

The notion of fd capsets works only for the class of fdcompact subsets To

treat other classes of subsets we need to extend this notion

Denition A class C of spaces is said to b e

i top ological if D C C implies D C

ii additiveif C C whenever C A B A and B are closed subsets of C and

A B C

iii closed hereditary if D C whenever D is a closed subset of a space C C

The nonambient case M Bestvina J Mogilski

Let C b e a class of spaces

Denition A subset A of an smanifold X is said to b e a C absorbing set of X if

i A has the hn complementin X

ii A is strongly C universal

iii A A where each A is a Z set of A and A C

n n n

Theorem Suppose a class C is topological additive and closedhereditary If

A and B are two C absorbing sets in an smanifold X then every open cover U

of X there exists a homeomorphism h X Y which is U close to the inclusion

A X

In general h cannot b e extended to anyambient homeomorphism of X

The ambient case J BaarsH GladdinesJ van Mill R Cauty T

Yagasaki etal

Let M b e a class of l tuples We assume that M is top ological additive

and closed hereditary We consider the following condition I

The condition I

I X X X isstronglyMuniversal

I there exist Z sets Z n of X suchthat

i X Z and ii Z Z X Z X M n

n n n n n l

In this case wehaveambient homeomorphisms

Theorem Suppose E is an smanifold and l tuples E X

X and E Y Y satisfy the condition I Then for any open cover U of E

l l

there exists a homeomorphism f E X X E Y Y with f id

l l E

TATSUHIKOYAGASAKI

Homotopyinvariance

Classication of innitedimensional manifolds is rather simple Qmanifolds are

classied by simple homotopy equivalence TA Chapman and smanifolds

are classied by homotopy equivalence D W Henderson and R M Schori

Y i X Y ho Theorem Suppose X and Y are smanifolds Then X

motopy equivalence

Characterization of infinitedimensional manifolds in term of

general position property and stability

Edwards program

There is a general metho d called as Edwards program of detecting top ological

E manifolds For innitedimensional top ological manifolds it takes the following

form Let X b e an ANR

i Construct a ne homotopy equivalence from an E manifold to the target X

ii Showthat f can b e approximated by homeomorphisms under some general

p osition prop ert yofX

This program yields basic characterizations of Qmanifolds smanifolds and

other incomplete manifolds

The complete cases

Qmanifolds

Theorem Aspace X is an Qmanifold i

i X is a local ly compact separable metrizable ANR

ii X has the disjoint cel ls property

smanifolds

Theorem Aspace X is an smanifold i

i X is a separable completely metrizable ANR

ii X has the strong discrete approximation property

Since the Qstability implies the disjoint cells prop erty and the sstability implies

the strong discrete approximation prop ertywe can replace the condition ii by

ii X is Qstable resp ectively sstable

The incomplete cases

BestvinaJ Mogilski has shown that in the incomplete case the ab ove M

program is formulated in the following form

INFINITEDIMENSIONAL MANIFOLD TUPLES

Theorem M BestvinaJ Mogilski

Suppose C is a class of spaces which is topological additive and closed hereditary

i For every ANR X thereexistsansmanifold M such that for every C absorbing

set in M there exists a ne homotopy equivalence f X

ii Suppose a X is a strongly C universal ANR and b X X where

each X is a strong Z set in X and X C Then every ne homotopy equivalence

i i

f X from any C absorbing set in an smanifold can be approximated by

homeomorphisms

Example manifolds and manifolds

Let C C denote the class of all nite dimensional compacta

c fdc

Theorem M BestvinaJ Mogilski

Aspace X is a manifold manifold i

i X is a separable ANR and compact fd compact

ii X is strongly C universal strongly C universal

c fdc

iii X X whereeach X is a strong Z set in X

n n

The condition iii can b e replaced by

iii X satises the strong discrete approximation prop erty

In HTorunczyk has obtained a characterization of manifolds in term of

stability

Theorem

X is a manifold i X is i a separable ANR ii fdcompact and iii

stable

Characterizations of s S S manifolds

In this section wewillinvestigate the problem of detecting s S S mani

folds Since we have obtained a characterization of smanifolds Theorem

the remaining problem is how to compare a tuple X X X lo cally with

s S S when X is an smanifold For this purp ose we will use the uniqueness

prop erty of absorbing sets in smanifolds x Since smanifolds are homotopy

invariant Theorem at the same time we can show the homotopyinvariance

of s S S manifolds

Characterizations of manifold tuples in term of the absorbing sets

Characterizations in term of capsets and fdcapsets

Theorem TA Chapman

X X is an s manifold s manifold i

i Xisan smanifold

ii X is a capset afdcapset

TATSUHIKOYAGASAKI

Suppose X X and Y Y are s manifolds s manifolds Then

Y Y i X Y X X

Theorem K SakaiRY Wong

X X X is an s manifold i

i X is an smanifold

ii X X is a cap fd cappair in X

Suppose X X X and Y Y Y are s manifolds Then X X X

Y Y Y i X Y

Characterizations in term of strong universality

We assume that s S S satises the condition I in Section

Theorem X X X is an s S S manifold i

l l

i X is an smanifold

ii X X X satises the condition I

Suppose X X X and Y Y Y are s S S manifolds Then

l l l

Y Y Y i X Y X X X

l l

Characterization in term of stabilityand homotopy negligible com

plement

General characterization theorem

We can show that the stability hn complement implies the strong univer

sality This leads to a characterization based up on the stability condition We

consider the following condition I I

The condition I I

I I S is contained in a countable union of Z sets of s

I I S has the hn complementins

I I Innite co ordinates There exists a sequence of disjoint innite subsets A

S N n such that for each i l and n a S S

i i A i

NnA

and b s S s S S S

A A A l l

n n n

Here for a subset A of N s denotes the pro jection onto the

k A

Afactor of s

Assumption We assume that s S S satises the condition I I

Notation Let MMs S S denote the class of l tuples X X

X which admits a layer preserving closed emb edding h X X X

l l

s S S

Theorem TYagasaki RCautyet al

Suppose Y Y Y satises the fol lowing conditions

i Y is a completely metrizable ANR

INFINITEDIMENSIONAL MANIFOLD TUPLES

ii Y has the hn complement in Y

iii Y Y Y is s S S stable

l l

Then Y Y Y is strongly Ms S S universal

l l

From Theorems wehave

Theorem X X X is an s S S manifold i

l l

i X is a completely metrizable ANR

ii X X X Ms S S

l l

iii X has the hn complement in X

iv X X X is s S S stable

l l

Suppose X X X and Y Y Y are s S S manifolds Then

l l l

X X X Y Y Y i X Y

l l

Examples

To apply Theorem wemust distinguish the class Ms S S This can

b e done for the triples s s s s and s

f f

This leads to the practical characterizations of manifolds mo deled on these triples

Ms the class of triples X X X suchthat

a X is completely metrizable b X is compact and c X is fdcompact

Theorem

X X X is an s manifold i

i X is a separable completely metrizable ANR

ii X is compact X is fdcompact

iii X has the hn complement in X

iv X X X is s stable

s s

Ms s the class of triples X X X such that

a X is completely metrizable b X is F in X c X is fdcompact

Theorem

X X X is an s s manifold i

i X is a separable completely metrizable ANR

subset of X X is fdcompact ii X is an F

iii X has the hn complement in X

iv X X X is s s stable

s f

TATSUHIKOYAGASAKI

Ms the class of triples X X X suchthat

a X is completely metrizable b X is F in X c X is fdcompact

Theorem

X X X is an s manifold i

i X is a separable completely metrizable ANR

ii X is an F subset of X X is fdcompact

iii X has the hn complement in X

iv X X X is s stable

Ms the class of triples X X X suchthat

a X is completely metrizable b X is F in X c X is compact

Theorem

X X X is an s manifold i

i X is a separable completely metrizable ANR

ii X is an F subset of X X is compact

iii X has the hn complement in X

stable iv X X X is s

In the next chapter these characterizations will b e applied to determine the lo cal

top ological typ es of some triples of homeomorphism groups of manifolds

Chapter Applications to homeomorphism groups of manifolds

Main problems

Notation

i H X denotes the homeomorphism group of a space of X with the compactop en

top ology

LIP

ii When X has a xed metric H X denotes the subgroup of lo cally LIP

homeomorphisms of X

iii When X is a p olyhedron H X denotes the subgroup of PLhomeomorphisms

of X

We shall consider the following problem

Problem

LIP

Determine the lo cal and global top ological typ es of groups H M H M

PL PL LIP PL

H M etc and tuples H M H M H M H M H M etc

INFINITEDIMENSIONAL MANIFOLD TUPLES

In the analogy with dieomorphism groups when X is a top ological manifold

we can exp ect that these groups are top ological manifold mo deled on some typical

innitedimensional spaces In fact RD Anderson showed that

Facts

i H R s

ii If G is a nite graph then H Gisansmanifold

After this result it was conjectured that

Conjecture H M isansmanifold for any compact manifold M

This basic conjecture is still op en for n and this imp oses a large restriction

to our work since most results in Chapter works only when ambient spaces are

smanifolds Thus in the present situation in order to obtain some results in

dimension n wemust assume that H M isansmanifold On the other hand

in dimension or we can obtain concrete results due to the following fact

Theorem R Luke WK Mason W Jakobsche

If X is a or dimensional compact polyhedron then H X is an smanifold

Belowwe shall follow the next conventions For a pair X A let H X Aff

H X f AAg When X A is a p olyhedral pair let H X AH X A

H X andH X PLA ff HX A f is PL on Ag The sup erscript c

denotes compact supp orts the subscript means orientation preserving

and denotes the identity connected comp onents of the corresp onding groups

An Euclidean PLmanifold means a PLmanifold which is a subp olyhedron of some

n n

R and has the standard metric induced from R Euclidean space

Stability properties of homeomorphism groups of polyhedra

First we shall summarize the stability prop erties of various triples of homeo

morphism groups of p olyhedra These prop erties will be used to determine the

corresp onding mo del spaces

Basic cases R Geoghegan J KeeslingD Wilson K Sakai

RY Wong

i If X is a top ological manifold then H X is sstable

ii If X is a lo cally compact p olyhedron then the pair H X H X is s

stable

iii If X is a Euclidean p olyhedron with the standard metric then the triple

PL LIP

H X is s stable H X H X

iv T Yagasaki If X K is a lo cally compact p olyhedral pair such that

dim K and dim X n K then H X K H X PLK H X K is

s s stable

TATSUHIKOYAGASAKI

Noncompact cases T Yagasaki

i If X is a noncompact lo cally compact p olyhedron then the triple H X H X

PLc

H X is s stable

ii If X is a noncompact Euclidean p olyhedron with the standard metric then the

LIP LIPc

triple H X H X H X is s stable

We can also consider the spaces of emb eddings Supp ose X and Y are Euclidean

p olyhedra Let E X Y denote the spaces of emb eddings of X into Y with the

LIP PL

compactop en top ology and let E X Y and E X Y denote the subspaces

of lo cally Lipschitz emb eddings and PLemb eddings resp ectively

Emb edding case K SakaiRY Wong cf

LIP PL

The triple E X Y E X Y E X Y is s stable

These stability prop erty are veried by using the Morse length of the image of

a xed segment under the homeomorphisms

LIP PL

The triple H M H X H X

H M

Supp ose M is a compact ndimensional manifold Since H M issstable by

the characterization of smanifold Theorem H M is an smanifold i it is

an ANR Here we face with the diculty of detecting innitedimensional ANRs

AV Cernavskii and RD Edwards RC Kirbyhaveshown

Theorem Lo cal contractibility H M is local ly contractible

H M

Supp ose M is a compact ndimensional PLmanifold

Basic Facts

J KeeslingD Wilson H M H M is s stable

D B Gauld H M islocallycontractible

R Geoghegan H M is fdcompact

i it is WE Haver A countable dimensional metric space is an ANR

lo cally contractible

From it follows that H M isalways an ANR Hence bythechar

acterization of manifold Theorem wehave

Main Theorem J KeeslingD Wilson H M is an manifold

Let H M cl H M Consider the condition

n and M for n

Under this condition H M is the union of some comp onents of H M

INFINITEDIMENSIONAL MANIFOLD TUPLES

Theorem R Geoghegan W E Haver

If H X is an smanifold and M satises thenH X H X is an s

manifold

LIP

H M K SakaiRY Wong

Supp ose M is a compact ndimensional Euclidean PLmanifold

Basic Facts

LIP

H M H M is s stable

LIP

H M is compact

Theorem

LIP

If H X is an smanifold and M satises thenH X H X is an s

manifold

LIP LIP

The triple H X H X H M T Yagasaki

Supp ose M is a compact ndimensional Euclidean PLmanifold

Basic Facts

LIP PL

K SakaiR Y Wong H M H M H M is s stable

LIP LIP PL

Let H X H X cl H M From Theorem Basic Facts and the

characterization of s manifolds Theorem it follows that

Theorem

LIP PL

If H X is an smanifold and M satises thenH X H X H X

is an s manifold

If X is a or dimensional compact Euclidean polyhedron with the standard

LIP PL

metric then H X H X H X is s manifold

Other triples

The triple H X K H X PLK H X K T Yagasaki

Theorem

i Suppose M is a compact PL nmanifold with M If n n

and H M is an smanifold then H M H M PLM H M is an s s

manifold

ii Suppose X K is a compact polyhedral pair such that dim X dim K

Then H X K H X PLK H X K is an s s and dim X n K

manifold

PL PLc LIP LIPc

The triples H X H X H X and H X H X H X T

Yagasaki

c PL PL

R dim case H R H R H s

dim case

TATSUHIKOYAGASAKI

Theorem If M is a noncompact connected PL manifold then H M

PL PLc

H M H M is an s manifold

Corollary

PL PL c

i If M R S R S P R n ptthenH M H M H M

S s

PL PLc

s ii In the remaining cases H M H M H M

There exist a LIPversion of the PL case

The group of quasiconformal QChomeomorphisms of a Riemann surface T

Yagasaki

Supp ose M is a connected Riemann surface Let H M denote the subgroup

of QChomeomorphisms of M

Theorem

i If M is compact then H M H M is an s manifold

ii If M is noncompact then H M H M is an s manifold

The space of emb eddings T Yagasaki

Supp ose M is a Euclidean PL manifold

LIP

Theorem If X is a compact subpolyhedron of M thenE X M E X M

E X M is an s manifold

Example The case X I

LIP PL

E I M E I M E I M S TM s

where S TM is the sphere bundle of the tangent bundle of M with resp ect to some

Riemannian metric

References

Anderson R D Hilb ert space is homeomorphic to the countable innite pro duct of lines

Bull Amer Math So c

Spaces of homeomorphisms of nite graphs unpublished manuscript

Baars J Gladdines H and van Mill J Absorbing systems in innitedimensional manifolds

Top ology and its Appl

Bessaga C and Pelczynski A Selected Topics in InniteDimensional Topology PWN

Warsaw

Bestvina M and Mogilski J Characterizing certain incomplete innitedimensional retracts

Michigan Math J

Cauty R Sur deux espaces de fonctions non derivables Fund Math

Dobrowolski T and Marciszewski W A contribution to the top ological classica

tion of the spaces C X Fund Math

Cernavskii A V Lo cal contractibility of the homeomorphism group of a manifold Dokl

Akad Nauk SSSR Soviet Math Dokl MR

INFINITEDIMENSIONAL MANIFOLD TUPLES

Chapman T A Dense sigmacompact subsets of innitedimensional manifolds Trans

Amer Math So c

Chapman T A Lectures on Hilbert Cube Manifolds CBMS American Mathematical

So cietyProvidence RI

Chigogidze A Inverse spectra NorthHolland Amsterdam

Edwards R D and Kirby R C Deformations of spaces of emb eddings Ann Math

Gauld D B Lo cal contractibility of spaces of homeomorphisms Comp ositio Math

Geoghegan R On spaces of homeomorphisms emb eddings and functions I Top ology

Geoghegan R On spaces of homeomorphisms emb eddings and functions I I The piecewise

linear case Pro c London Math So c

Geoghegan R and Haver W E On the space of piecewise linear homeomorphisms of a

manifold Pro c Amer Math So c

Haver W E Lo cally contractible spaces that are absolute neighborhood retracts Pro c

Amer Math So c

Henderson D W and Schori R M Top ological classication of innitedimensional mani

folds by homotopytyp e Bull Amer Math So c

Jakobsche W The space of homeomorphisms of a dimensional p olyhedron is an

manifold Bull Acad Polon Sci Ser Sci Math

Keesling J Using ows to construct Hilb ert space factors of function spaces Trans Amer

Math So c

and Wilson D The group of PLhomeomorphisms of a compact PLmanifold is an

manifold Trans Amer Math So c

Luke R and Mason W K The space of homeomorphisms on a compact two manifold is

an absolute neighb orho o d retract Trans Amer Math So c

Mogilski J Characterizing the top ology of innitedimensional compact manifolds Pro c

Amer Math So c

van Mill J InniteDimensional Topology Prerequisites and Introduction NorthHolland

Amsterdam

Schori R M Top ological stability for innitedimensional manifolds Comp ositio Math

Sakai K and Wong R Y On the space of Lipschitz homeomorphisms of compact p olyhe

dron Pacic J Math

and Wong R Y On innitedimensional manifold triples Trans Amer Math So c

Torunczyk H Absolute retracts as factors of normed linear spaces Fund Math

On Cartesian factors and the top ological classication of linear metric spaces Fund

Math

Characterizing Hilb ert space top ology Fund Math

West J E The ambient homeomorphy of an incomplete subspace of innitedimensional

Hilb ert spaces Pacic J Math

Yagasaki T Innitedimensional manifold triples of homeomorphism groups Top ology

Appl to app ear

Homeomorphism groups of noncompact manifolds in preparation

The groups of quasiconformal homeomorphisms on Riemann surfaces preprint

Department of Mathematics Kyoto Institute of Technology Matsugasaki Sakyo

ku Kyoto Japan

Email address yagasakiipckitacjp