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Home , Alexander duality

ISSN 1364-0380 51

Geometry

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olume V

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Published Octob er

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Duality Grop es and Alexander

acheslav S Krushkal Vy

Peter Teichner

Department of Mathematics Michigan State University

East Lansing MI USA

Current address MaxPlanckInstitut fur Mathematik

D Bonn Germany GottfriedClarenStrasse

and

Department of Mathematics

University of California in San Diego

La Jolla CA USA

Email krushkalmathmsuedu and teichnereucliducsdedu

Abstract

We prove a geometric renement of Alexander for certain complexes

the socalled gropes emb edded into space This renement can b e roughly

formulated as saying that dimensional Alexander duality preserves the dis

joint Dwyer ltration

In addition we give new pro ofs and extended versions of two lemmas of Freed

which are of central imp ortance in the ABslice problem the man and Lin

main op en problem in the classication theory of top ological Our

metho ds are group theoretical rather than using Massey pro ducts and Milnor

invariants as in the original pro ofs

Classication numb ers Primary M M AMS

Secondary M N N

ords Alexander duality manifolds grop es link homotopy Milnor Keyw

group Dwy er ltration

osed Robion Kirby Received June Prop

Michael Freedman Ronald Stern Revised Octob er Seconded

Copyright Geometry and Topology

52 Vyacheslav S Krushkal and Peter Teichner

Intro duction

n

Consider a nite complex X PLemb edded into the n dimensional S

n

integer H S r X with Alexander duality identies the reduced

i

1i n

X This implies that the homology or even the sta the H

ble homotopy typ e of the complement cannot distinguish b etween p ossibly

n

t emb eddings of X into S Note that there cannot b e a duality for ho dieren

motopy groups as one can see by considering the fundamental group of classical

1

complements ie the case X S and n

Ho wever one can still ask whether additional information ab out X do es lead to

n

additional information ab out S r X For example if X is a smo oth closed n

i dimensional then the cohomological fundamental class is dual

n

to a spheric al class in H S r X Namely it is represented by any meridional

i

i sphere which by denition is the b oundary of a normal disk at a p oint in X

This geometric picture explains the dimension shift in the Alexander duality

theorem

n

By reversing the roles of X and S r X in this example we see that it is not

n

true that H X b eing spherical implies that H S r X is spherical

i n1i

However the following result shows that there is some kind of improved duality

es in if one do es not consider linking dimensions One should think of the Grop

our theorem as means of measuring how spherical a homology class is

4

Grop e Duality If X S is the disjoint union of closed em Theorem

4

b edded Grop es of class k then H S r X is freely generated by r disjointly

2

Here r is the rank of H X Moreover emb edded closed Grop es of class k

1

4

H S r X cannot b e generated by r disjoint maps of closed grop es of class

2

k

As a corollary to this result we show in that certain Milnor invariants of

3

a link in S are unchanged under a Grope concordance

The Gropes ab ove are framed thickenings of very simple complexes called

gropes which are inductively built out of surface stages see Figure and Sec

a surface with a single b oundary tion For example a grop e of class is just

comp onent and grop es of bigger class contain information ab out the lower cen

tral series of the fundamental group Moreover every closed grop e has a fun

damental class in H X and one obtains a geometric denition of the Dwyer

2

ltration

X X X X H X

2 3 2 2 k

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 53

by dening X to b e the set of all homology classes represented by maps

k

of closed grop es of class k into X Theorem can thus b e roughly formulated

as saying that dimensional Alexander duality preserves the disjoint Dwyer

ltration

Figure A grop e of class is a surface two closed grop es of class

Figure shows that each grop e has a certain typ e which measures how the

surface stages are attached In Section this will b e made precise using certain

ro oted trees compare Figure In Section we give a simple algorithm for

obtaining the trees corresp onding to the dual Grop es constructed in Theorem

The simplest application of Theorem with class k is as follows Consider

2 4

the standard emb edding of the torus T into S which factors through the

2 3

usual unknotted picture of T in S Then the b oundary of the normal bundle

2

of T restricted to the two essential circles gives two disjointly emb edded tori

2 4 2

Z Since b oth of these tori may b e representing generators of H S r T

2

4 2

surgered to emb edded H S r T is in fact spherical However

2

it cannot b e generated by two maps of spheres with disjoint images since a

map of a sphere may b e replaced by a map of a grop e of arbitrarily big class

This issue of disjointness leads us to study the relation of grop es to classical link

homotopy We use Milnor group techniques to give new pro ofs and improved

namely the Grope Lemma and the Link versions of the two central results of

Comp osition Lemma Our generalization of the grop e lemma reads as follows

3

Two n comp onent links in S are link homotopic if and only if Theorem

3

I they cob ound disjointly immersed annuluslike grop es of class n in S

This result is stronger than the version given in where the authors only make

a comparison with the trivial link Moreover our new pro of is considerably

shorter than the original one

Geometry and Topology, Volume 1 (1997)

54 Vyacheslav S Krushkal and Peter Teichner

Our generalization of the link comp osition lemma is formulated as Theorem

in Section The reader should b e cautious ab out the pro of given in It

turns out that our Milnor group approach contributes a b eautiful feature to

algebraization of link homotopy He proved in that by forgetting Milnors

one comp onent of the unlink one gets an ab elian normal subgroup of the Milnor

group which is the additive group of a certain ring R We observe that the

Magnus expansion of the free Milnor groups arises naturally from considering

the conjugation action of the quotient group on this ring R Moreover we

show in Lemma that comp osing one link into another corresp onds to

multiplication in that particular ring R This fact is the key in our pro of of

the link comp osition lemma

Our pro ofs completely avoid the use of Massey pro ducts and Milnor invariants

and we feel that they are more geometric and elementary than the original

pro ofs This might b e of some use in studying the still unsolved ABslice

problem which is the main motivation b ehind trying to relate grop es their du

ality and link homotopy It is one form of the question whether top ological

surgery and scob ordism theorems hold in dimension without fundamental

group restrictions See for new developments in that area

It is a pleasure to thank Mike Freedman for many imp or Acknow ledgements

tant discussions and for providing an inspiring atmosphere in his seminars In

particular we would like to p oint out that the main construction of Theorem

is reminiscent of the metho ds used in the linear grope height raising pro cedure

of The second author would like to thank the Miller foundation at UC

Berkeley for their supp ort

Preliminary facts ab out grop es and the lower

central series

The following denitions are taken from

A grope is a sp ecial pair complex circle A grop e has a Denition

class k For k a grop e is dened to b e the pair circle

with a circle For k a grop e is precisely a compact oriented surface

single b oundary comp onent For k nite a k grope is dened inductively as

i genusg follow Let f b e a standard symplectic basis of circles

i i

k and p q k for for For any p ositive integers p q with p q

i i i i i i

0 0

at least one index i a k grop e is formed by gluing p grop es to each and

0 i i

q grop es to each

i i

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 55

The imp ortant information ab out the branching of a grop e can b e very well

captured in a ro oted tree as follows For k this tree consists of a single

which is called the root For k one adds vertex v genus edges

0

to v and may lab el the new vertices by Inductively one gets the

0 i i

tree for a k grop e which is obtained by attaching p grop es to and q

i i i

grop es to by identifying the ro ots of the p resp ectively q grop es with

i i i

the vertices lab eled by resp ectively Figure b elow should explain the

i i

corresp ondence b etween grop es and trees

leaves

ro ot

Figure A grop e of class and the asso ciated tree

Note that the vertices of the tree which are ab ove the ro ot v come in pairs

0

corresp onding to the symplectic pairs of circles in a surface stage and that such

ro oted paired trees corresp ond bijectively to grop es Under this bijection the

valent vertices of the tree corresp ond to circles on the grop e which leaves

freely generate its fundamental group We will sometimes refer to these circles

as the tips of the grop e The b oundary of the rst stage surface will b e

referred to as the bottom of the grop e

k

Given a group we will denote by the k th term in the lower central series

1 k k 1

of dened inductively by and the characteristic

subgroup of k fold commutators in

Lemma Algebraic interpretation of grop es For a space X a lo op

k

lies in X k if and only if b ounds a map of some k grop e

1

k

Moreover the class of a grop e G is the maximal k such that G

1

2

A closed k grop e is a complex made by replacing a cell in S with a k

grop e A closed grop e is sometimes also called a spherelike grop e Similarly

Geometry and Topology, Volume 1 (1997)

56 Vyacheslav S Krushkal and Peter Teichner

one has annuluslike k grop es which are obtained from an annulus by replacing

a cell with a k grop e Given a space X the Dwyers subgroup X of

k

H X is the set of all homology classes represented by maps of closed grop es of

2

class k into X Compare for a translation to Dwyers original denition

k b e a p ositive integer and let f Theorem Dwyers Theorem Let

X Y b e a map inducing an on H and an epimorphism on

1

k

Then f induces an isomorphism on H

1 1 2

k

A Grope is a sp ecial untwisted dimensional thickening of a grop e G

it has a preferred around the base circle in its b oundary This

3

and taking untwisted thickening is obtained by rst emb edding G in R

its thickening there and then crossing it with the interval The deni

3

tion of a Grop e is indep endent of the chosen emb edding of G in R One

can alternatively dene it by a thickening of G such that all relevant relative

numb ers vanish Similarly one denes sphere and annuluslike Grop es Euler

the capital letter indicating that one should take a dimensional untwisted

thickening of the corresp onding complex

The Grop e Lemma

0

and L in We rst recall some material from Two n comp onent links L

3

S are said to b e linkhomotopic if they are connected by a parameter family

of immersions such that distinct comp onents stay disjoint at all times L is said

to b e homotopical ly trivial if it is linkhomotopic to the unlink L is almost

homotopical ly trivial if each prop er sublink of L is homotopically trivial

with For a group normally generated by g g its Milnor group M

1

k

resp ect to g g is dened to b e the quotient of by the normal subgroup

1

k

h

generated by the elements g g where h is arbitrary Here we use the

i

i

conventions

1 1

h 1

h g h g g g g g g and g

1 2 1 2

1 2

k +1

M is nilp otent of class k ie it is a quotient of and is generated

by the quotient images of g g see The Milnor group M L of a link

1

k

3

It is the L is dened to b e M S r L with resp ect to its meridians m

i 1

largest common quotient of the fundamental groups of all links linkhomotopic

to L hence one obtains

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 57

0

Theorem Invariance under link homotopy If L and L are link homo

topic then their Milnor groups are isomorphic

3

The track of a link homotopy in S I gives disjointly immersed annuli with

the additional prop erty of b eing mapp ed in a level preserving way However

0

this is not really necessary for L and L to b e link homotopic as the following

result shows

Lemma Singular concordance implies homotopy

3 0 3 3

If L S fg and L S fg are connected in S I by disjointly

0

immersed annuli then L and L are linkhomotopic

Remark This result was recently generalized to all dimensions see

Our Grop e Lemma Theorem in the intro duction further weakens the con

0

ditions on the ob jects that connect L and L

of of Theorem Let G G b e disjointly immersed annuluslike Pro

1 n

0 3

grop es of class n connecting L and L in S I To apply the ab ove Lemma

in the com we want to replace one G at a time by an immersed annulus A

i i

plement of all grop es and annuli previously constructed

Lets start with G Consider the circle c which consists of the union of the

1 1

0

then the rst of L then an arc in G leading from l to l rst comp onent l

1 1 1

1

0 0

of L and nally the same arc back to the base p oint Then the comp onent l

1

n grop e G b ounds c and thus c lies in the n th term of the lower central

1 1 1

3

series of the group S I r G where G denotes the union of G G

1 2 n

As rst observed by Casson one may do nitely many nger moves on the

b ottom stage surfaces of G keeping the comp onents G disjoint such that the

i

natural pro jection induces an isomorphism

3 3

S I r G M S I r G

1 1

h

see for the precise argument the key idea b eing that the relation m m

i

i

can b e achieved by a self nger move on G which follows the lo op h But

i

the latter Milnor group is normally generated by n meridians and is thus

3

nilp otent of class n In particular c b ounds a disk in S I r G which

i

0

cob ound an annulus A disjoint from is equivalent to saying that l and l

1 1

1

G G

2 n

Since nger moves only change the immersions and not the typ e of a complex

ie an immersed annulus stays an immersed annulus the ab ove argument can

b e rep eated n times to get disjointly immersed annuli A A connecting

1 n

0

L and L

Geometry and Topology, Volume 1 (1997)

58 Vyacheslav S Krushkal and Peter Teichner

Grop e Duality

In this section we give the pro of of Theorem and a renement which explains

what the trees corresp onding to the dual Grop es lo ok like Since we now con

sider closed grop es the following variation of the corresp ondence to trees turns

0

out to b e extremely useful Let G b e a closed grop e and let G denote G with

to b e the a small cell removed from its b ottom stage We dene the tree T

G

0

tree corresp onding to G as dened in Section together with an edge added

to the ro ot vertex This edge represents the deleted cell and it turns out to

b e useful to dene the ro ot of T to b e the valent vertex of this new edge

G

See Figure for an example of such a tree

Pro of of Theorem Abusing notation we denote by X the core grop e of

4

the given dimensional Grop e in S Thus X is a complex which has a

4

particularly simple thickening in S which we may use as a regular neighb or

ho o d All constructions will take place in this regular neighb orho o d so we may

g denote a assume that X has just one connected comp onent Let f

ij ij

standard symplectic basis of curves for the i th stage X of X these curves

i

corresp ond to vertices at a distance i from the ro ot in the asso ciated tree

Here X is the b ottom stage and thus a closed connected surface For i

1

the X are disjoint unions of punctured surfaces They are attached along some

i

of the curves or

i1j i1j

4

Let A denote the circle bundle of X in S restricted to a parallel dis

ij i

placement of in X see Figure The corresp onding disk bundle for

ij i

small enough can b e used to see that the torus A has linking numb er

ij

g Note with and do es not link other curves in the collection f

st ij st

that if there is a higher stage attached to then it intersects A in a single

ij ij

p oint while if there is no stage attached to then A is disjoint from X

ij ij

4

and the generator of H S r X represented by A is Alexanderdual to

2 ij ij

denote a torus representative of the class dual to Similarly let B

ij ij

There are two inductive steps used in the construction of the stages of dual

Grop es

0

Let b e a curve in the collection f g and let X denote the Step

ij ij

subgrop e of X which is attached to Since X is framed and emb edded a

4 0

parallel copy of in S b ounds a parallel copy of X in the complement of X

If there is no higher stage attached to then the application of Step to this

curve is empty

Let b e a connected comp onent of the i th stage of X and let m Step

i i

4

denote a meridian of in S that is m is the b oundary of a small normal

i i

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 59

i

m

i

ij



i1n i1

ij

0

X

m A

i1n ij i1

i

displaced

ij

Steps and Figure

disk to at an interior p oint Supp ose i and let denote the previous

i i1

stage so that is attached to along some curve say The torus

i i1 i1n

B meets in a p oint but making a puncture into B around this

i1n i i1n

intersection p oint and connecting it by a tub e with m exhibits m as the

i i

b oundary of a punctured torus in the complement of X see Figure

By construction H X is generated by those curves f g which do not

1 ij ij

have a higher stage attached to them Fix one of these curves say We will

ij

4

show that its dual torus A is the rst stage of an emb edded Grop e G S r X

ij

of class k The meridian m and a parallel copy of form a symplectic basis

i ij

of circles for A Apply Step to If i the result of Step is a grop e

ij ij

at least of class k and we are done If i apply in addition Step to m

i

The result of Step is a grop e with a new genus surface stage the tips of

which are the meridian m and a parallel copy of one of the standard curves

i1

in the previous stage say The next Step Step cycle is applied to

i1n

these tips Altogether there are i cycles forming the grop e G

The trees corresp onding to dual grop es constructed ab ove may b e read o the

for X and pick the tip tree asso ciated to X as follows Start with the tree T

X

valent vertex corresp onding to the curve The algorithm for drawing

ij

the tree T of the grop e G Alexanderdual to reects Steps and ab ove

G ij

Consider the path p from to the ro ot of T and start at the vertex

ij X i1n

adjacent to Erase all branches in T growing from except for

ij X i1n

the edge which has b een previously considered and its partner

ij i1n

branch and then move one edge down along the path p This step

ij i1n

Geometry and Topology, Volume 1 (1997)

60 Vyacheslav S Krushkal and Peter Teichner

is rep eated i times until the ro ot of T is reached The tree T is obtained

X G

by copying the part of T which is not erased with the tip drawn as the

X ij

ro ot see gure

m

ij

ij 1

m

2

i1n

T T

G X

Figure A dual tree The branches in T to b e erased are drawn with dashed lines

X

Note the distinguished path in T starting at the ro ot and lab elled by

G

m m m Each of the vertices m m m is trivalent this cor

i i1 1 i i1 2

resp onds to the fact that all surfaces constructed by applications of Step have

genus see gures In particular the class of G may b e computed as the

sum of classes of the grop es attached to the partner vertices of m m

i 1

plus

We will now prove that the dual grop e G is at least of class k The pro of is

by induction on the class of X For surfaces class the construction gives

tori in the collection fA B g Supp ose the statement holds for Grop es of

ij ij

class less than k and let X b e a Grop e of class k By denition for each

standard pair of dual circles in the rst stage of X there is a p grop e

attached to and a q grop e X attached to with p q k Let b e X

one of the tips of X By the induction hyp othesis the grop e G dual to

given by the construction ab ove for X is at least of class p G is obtained

from G by rst attaching a genus surface to m with new tips m and a

2 1

parallel copy of Step and then attaching a parallel copy of X Step

According to the computation ab ove of the class of G in terms of its tree it is

equal to p q k

It remains to show that the dual grop es can b e made disjoint and that they

are framed Each dual grop e may b e arranged to lie in the b oundary of a

regular neighb orho o d of X for some small Figure shows how Steps

and are p erformed at a distance from X Note that although tori A and ij

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 61

B intersect at most one of them is used in the construction of a dual grop e

ij

Taking distinct values the grop es are arranged for each index i j

1 r

to b e pairwise disjoint The same argument shows that each grop e G has a

0 0 4

parallel copy G with G G hence its thickening in S is standard

m

i

0

a parallel X

0

copy of X

ij



ij i

A

ij

displaced

ij

punctured B

ij

Figure Steps and at a distance from X

4

To prove the converse part of the theorem supp ose that H S r X is gener

2

ated by r disjoint maps of closed grop es Perturb the maps in the complement

of X so that they are immersions and their images have at most a nite num

b er of transverse selfintersection p oints The usual pushing down and twisting

pro cedures from pro duce closed disjoint framed grop es G G whose

1 r

only failure to b eing actual Grop es lies in p ossible selfintersections of the b ot

tom stage surfaces The G still lie in the complement of X and have class k

i

The pro of of the rst part of Theorem shows that H Y Y is gener

2

k +1

ated by the Cliord tori in the neighb orho o ds of selfintersection p oints of the

4

G where Y denotes the complement of all G in S Assume X is connected

i i

0

otherwise consider a connected comp onent of X and let X denote X with

a cell removed from its b ottom stage The relations given by the Cliord

tori are among the dening relations of the Milnor group on meridians to the

grop es and Dwyers theorem shows as in Lemma that the inclusion

map induces an isomorphism

0 0 k +1 k +1

M X M X M Y M Y

1 1 1 1

0

Consider the b oundary curve of X Since X is a grop e of class k by

0 k +1

Lemma b elow we get M X On the other hand b ounds a

1

disk in Y hence M Y This contradiction concludes the pro of of

1

Theorem

k +1

Lemma Let G b e a grop e of class k Then M G 1

Geometry and Topology, Volume 1 (1997)

62 Vyacheslav S Krushkal and Peter Teichner

Pro of This is b est proven by an induction on k starting with the fact that

is freely generated by all and Here is the b ottom surface stage of

1 i i

the grop e G with a standard symplectic basis of circles The Magnus

i i

expansion for the free Milnor group see or the pro of of Theorem

Q

3

shows that do es not lie in M Similarly for k G

i i 1 1

is freely generated by those circles in a standard symplectic basis of a surface

stage in G to which nothing else is attached Now assume that the k grop e

to and q grop es G to G is obtained by attaching p grop es G

i i i

i

i

+1 q +1 p

i i

and M G p q k By induction M G

i 1 i i i i 1

i

i

since p q But the free generators of G and G are contained

i i 1 1

i

i

Q

k +1

in the set of free generators of G and therefore M G

1 i i 1

Again this may b e seen by applying the Magnus expansion to M G

1

Remark In the case when all stages of a Grop e X are tori the corresp ondence

b etween its tree T and the trees of the dual Grop es given in the pro of of

X

theorem is particularly app ealing and easy to describ e Let b e a tip of

T The tree for the Grop e Alexanderdual to is obtained by redrawing

X

T only with drawn as the ro ot see Figure

X

m

1

m

2

m

3

T T

X G

Figure Tree duality in the genus case

As a corollary of Theorem we get the following result

0 3 0 0

l b e two links in S Corollary Let L l l and L l

1 n

n

1

3

ose there are disjointly emb edded annulus fg and S fg resp ectively Supp

3 0

like Grop es A A of class k in S with A l l i n

1 n i i

i

ts Then there is an isomorphism of nilp otent quotien

3 3 k 3 0 3 0 k

S r L S r L S r L S r L

1 1 1 1

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 63

Remark For those readers who are familiar with Milnors invariants we

should mention that the ab ove statement directly implies that for any multi

0

index I of length jI j k one gets I I For a dierent pro of of this

L

L

consequence see

Pro of of Corollary The pro of is a version of Stallings pro of of the

k

3

concordance invariance of all nilp otent quotients of S r L see Namely

1

Alexander duality and Theorem imply that the inclusion maps

3 3 0 3

S r A A S fg r L S fg r L

1 n

induce on H and on H So by Dwyers Theorem

1 2

k

k

they induce isomorphisms on

1 1

The Link Comp osition Lemma

The Link Comp osition Lemma was originally formulated in The reader

should b e cautious ab out its pro of given there it can b e made precise using

ts while this section presents an Milnors invariants with rep eating co ecien

alternative pro of

3

b

Given a link L l l in S and a link Q q q in the solid

1 1 m

k +1

2 1

their comp osition is obtained by replacing the last comp onent torus S D

b

of L with Q More precisely it is dened as L Q where L l l

1

k

1 2 3

and S D S is a framed emb edding whose image is a tubular

2

neighb orho o d of l The meridian fg D of the solid torus will b e

k +1

b b

and we put Q Q We sometimes think of Q or Q as links denoted by

3 1 2 3

in S via the standard emb edding S D S

Theorem Link Comp osition Lemma

3

b b

i If L and Q are b oth homotopically essential in S then L Q is also

homotopically essential

b b

versely if L Q is homotopically essential and if b oth L and Q are ii Con

b b

almost homotopically trivial then b oth L and Q are homotopically essential

3

in S

Remark Part ii do es not hold without the almost triviality assumption on

b b b

L and Q For example let L consist of just one comp onent l and let Q b e

1

1 2

a Hopf link contained in a small ball in S D Then L Q Q is

b

homotopically essential yet L is trivial

Geometry and Topology, Volume 1 (1997)

64 Vyacheslav S Krushkal and Peter Teichner

0

^

L

^

1 2

S D 

b

Figure In this example L is the and Q is the Bing double of the

1 2

core circle of S  D

In part i if either L or Q is homotopically essential then their comp osition

b b

L Q is also essential Note that Q and Q are homotopically equivalent

see Lemma in If neither L nor Q is homotopically essential then by

b

deleting some comp onents of L and Q if necessary one may assume that L

b

and Q are almost homotopically trivial and still homotopically essential In

the case when L Q is not almost homotopically trivial part i follows

immediately Similarly part ii can b e proved in this case easily by induction

on the numb er of comp onents of L and Q

b b

efore we wil l assume from now on that L Q and L Q are almost Ther

3

homotopical ly trivial links in S

3

b b

If L and Q are b oth homotopically trivial in S then rep Lemma

resents the trivial element in the Milnor group M L Q

0 1

Pro of Let denote S fg The Milnor group M L Q is nilp otent

of class k m so it suces to show that represents an element in

3 k +m+1

S r L Q This will b e achieved by constructing an grop e

1

G b ounded by in the complement of L Q In fact the construction

0 3 0

also gives an grop e G S r L Q b ounded by

1 2 3

Consider S D as a standard unknotted solid torus in S and let c denote

2 1

b

the core of the complementary solid torus D S Since Q is homotopically

1 2

trivial after changing Q by an appropriate link homotopy in S D b ounds

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 65

3

an immersed disk S in the complement of the new link Denote the new

link by Q again Similarly L can b e changed so that the untwisted parallel

0 0 3

copy of l b ounds a disk S r L Recall that M L Q do es

k +1

not change if L Q is mo died by a link homotopy

The intersection numb er of with c is trivial since and c do not link

2 1 2 1

Replace the union of disks D S by annuli lying in D S to

1 2

get S D r Q an immersed surface b ounded by Similarly the

0 1 2 0

intersection numb er of with the core circle of S D is trivial and

0 3 0

b ounds S r L Q The surfaces and are the rst stages

0

of the grop es G and G resp ectively

0

Notice that half of the basis for H is represented by parallel copies of

1

0

They b ound the obvious surfaces annuli connecting them with union with

0

which provide the second stage for G Since this construction is symmetric

0

it provides all higher stages for b oth G and G

3 3

b

Lemma Let i S r neighb orho o d L r l S r L Q r l denote

1 1

the inclusion map and let i b e the induced map on Then i induces a

1

# #

map i of Milnor groups well dened



Remark Given two groups G and H normally generated by g resp ectively

i

h let MG and MH b e their Milnor groups dened with resp ect to the given

j

sets of normal generators If a homomorphism G H maps each g to

i

one of the h then it induces a homomorphism M MG MH In general

j

G H induces a homomorphism of the Milnor groups if and only if g

i

(g )

in MH for all i and all g G commutes with g

i

b

Pro of of Lemma The Milnor groups M L r l and M L Q r l

1 1

are generated by meridians Moreover i m m for i k and

i i

#

b

i m where m m are meridians to the comp onents of L

1

# k +1 k +1

Hence to show that i is welldened it suces to prove that all the commuta



tors

3 (g ) i

#

b

g S r L r l

1 1

are trivial in M L Q r l Consider the following exact sequence obtained

1

by deleting the comp onent q of Q

1

k er M L Q r l M L Q r l q

1 1 1

Geometry and Topology, Volume 1 (1997)

66 Vyacheslav S Krushkal and Peter Teichner

b b

An application of Lemma to L r l and to Q r q shows that

1 1

g

and hence k er The observation that k er is generated

by the meridians to q and hence is commutative nishes the pro of of

1

Lemma

Pro of of Theorem Let M F b e the Milnor group of a free group

m m

+1 1 s

comp onents with meridians m ie the Milnor group of the trivial link on s

i

Let R y y b e the quotient of the free asso ciative ring on generators

1 s

with one index y y y by the ideal generated by the monomials y

i 1 s i

r

1

o ccurring at least twice The additive group R y y of this ring is

1 s

y free ab elian on generators y where all indices are distinct Milnor

i i

r

1

showed that setting m induces a short exact sequence of groups

s+1

r i

R y y M F M F

1 s m m m m

s

1 s+1 1

where r is dened on the ab ove free generators by leftiterated commutators

with m

s+1

m m m m r y y

j j j s+1 j j

1 2 1

k k

In particular r and r m Obviously the ab ove extension

s+1

of groups splits by sending m to m This splitting induces the following

i i

conjugation action of M F then on R y y Let Y y y

m m 1 s j j

s

1 1

k

1

m r Y m m r Y r Y

i i

i

m r Y r y Y m m m m

s+1 i i j j j

1 2

k

which implies that m acts on R y y by ring multiplication with y

i 1 s i

on the left Since m generate the group M F this denes a well

i m m

s

1

dened homomorphism of M F into the units of the ring R y

m m 1

s

1

y In fact this is the Magnus expansion well known in the context of free

s

groups rather than free Milnor groups We conclude in particular that the

ab elian group R y y is generated by y as a mo dule over the group

1 s i

M F

m m

s

1

Returning to the notation of Theorem we have the following commutative

diagram of group extensions We use the fact that the links L Q r l and

1

b

L r l are homotopically trivial Here y are the variables corresp onding to the

1 i

are the variables corresp onding to Q We intro duce short link L and z

j

R y y and R Y Z R y y z z notations R Y

1 1 2 m

k k

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 67

r i

R Y Z M L Q r l M L Q r l q

1 1 1

x x x

c l l c

j

r 

b

M L r l M L r l R Y

1 1

Recall that by denition l cm m for all meridians m m of L r l

i i 2 1

k

Moreover the link comp osition map l c sends the meridian m to the curve

k +1

of Q

The existence of the homomorphism l c on the Milnor group level already implies

our claim ii in Theorem By assumption l represents the trivial element

1

b b

in M L r l since L is homotopically trivial Consequently l cl l is also

1 1 1

trivial in M L Q r l and hence by the link L Q is homotopically

1

trivial

The key fact in our approach to part i of Theorem is the following result

which says that link comp osition corresp onds to ring multiplication

R y y R y y z z Lemma The homomorphism

2 2 2 m

k k

1

is given by ring multiplication with r on the right

Note that by Lemma is trivial in M L Q r l q so that it

1 1

1

makes sense to consider r We will abbreviate this imp ortant element by

R

Pro of of Lemma Since the ab ove diagram commutes and R y y

2

k

is generated by y as a mo dule over the group M F it suces to

i m m

2

k

check our claim for these generators y We get by denition

i

1

1

l cr y l cm m m m m

i i i i

k +1

i

1

r y r y

i R i R

We are using the fact that conjugation by m corresp onds to left multiplication

i

by y

i

b

Since L is homotopically trivial and L is homotopically essential it follows

l k er j After p ossibly reordering the y this implies in addition that

1 i

that for some integer a we have

1

r l a y y terms obtained by p ermutations from y y

1 2 2

k k

Geometry and Topology, Volume 1 (1997)

68 Vyacheslav S Krushkal and Peter Teichner

Setting all the meridians m of L to which implies setting the variables y

i i

to we get a commutative diagram of group extensions

r i

R Z M Q M Q r q

1

x x x

p

r i

R Y Z M L Q r l M L Q r l q

1 1 1

As b efore R Z and R Y Z are short notations for R z z and R y

2 m 1

b b

y z z resp ectively Since Q and equivalently Q is homotopi

2 m

k

cally essential we have k er i This shows that p The almost

R

b

triviality of Q implies in addition that after p ossibly reordering the z we have

j

for some integer b

p b z z terms obtained by p ermutations from z z

R 2 m 2 m

1 1

It follows from Lemma that r l r l This pro duct contains

1 1 R

the term

ab y y z z

2 2 m

k

the co ecient ab of which is nonzero This completes the pro of of Theorem

Remark Those readers who are familiar with Milnors invariants will have

recognized that the ab ove pro of in fact shows that the rst nonvanishing

invariants are multiplicative under link comp osition

Geometry and Topology, Volume 1 (1997)

Alexander Duality, Gropes and Link Homotopy 69

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C Gien Link concordance implies link homotopy Math Scand

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Geometry and Topology, Volume 1 (1997)