The algebraic surgery exact
ANDREW RANICKI Edinburgh
http://www.maths.ed.ac.uk/ aar
e
algebraic surgery exact sequence
• The
de ned for any space is X
A
; [ ] Hn X L Ln Z π1 X
···→ • −→
; Sn X Hn 1 X L ...
→ → − • →
the -theory assembly map. The
with A L
functor is homotopy invariant. X S X
7→ ∗
2-stage obstructions of the Browder-
• The
Novikov-Sullivan-Wall surgery theory for the
man- existence and uniqueness of top ological
ifold structures in a homotopy typ e are re-
placed by single obstructions in the relative
groups of the assembly map . S X A
∗ 1
Lo cal and global mo dules
The assembly map : ; [ ] A H X L L Z π1 X
• ∗ • → ∗
is induced by a forgetful functor
: -mo dules [ ]-mo dules A Z,X Z π1 X
{ }→{ }
where the domain dep ends on the lo cal
and the target dep ends only of
top ology X
on the fundamental group of , which
π1 X X
is global.
!
with terms of sheaf theory = In A q p
• !
the universal covering projection p : X X
→
and pt. . q : X f →{ }
f
geometric mo del for the -theory as- The L
•
is the forgetful functor
sembly A
Poincar e complexes
{geometric } manifolds top ological .
→{ }
fact, in dimensions 5 this functor In n
≥
has the same bre as
A. 2
Lo cal and global quadratic Poincar e
complexes
Global The -group [ ] is the L Ln Z π1 X
•
rdism group of -dimensional quadratic
cob o n
Poincar e complexes over [ ].
C, ψ Z π1 X
The generalized homology group
• Lo cal
; is the cob ordism group of - Hn X L n
•
dimensional quadratic Poincar e complexes
over . As in sheaf theory
C, ψ Z,X C
stalks, which are -mo dule chain com-
has Z
. plexes C x x X
∈
is the cob ordism group Lo cal-Global Sn X
•
of quadratic Poincar e n 1 -dimensional
−
over such that the
complexes C, ψ Z,X
[ ]-mo dule chain complex assembly
Z π1 X
is acyclic.
A C 3
The total surgery obstruction
total surgery obstruction of an
• The
geometric Poincar e complex
n-dimensional
the cob ordism class
X is
= s X C, ψ Sn X
∈
of a [ ]-acyclic 1 -dimensional Z π1 X n
−
quadratic Poincar e complex over .
C, ψ Z,X
measure the fail- stalks The C x x X
∈
ure of have lo cal Poincar e duality X to
n r
Hr C x H x Hr X, X x ···→ → − { } → \{ }
n r +1
Hr 1 C x H − x ...
→ − → { } →
an -dimensional homology manifold if
X is n
= 0. In particular, this only if and H C x
∗
the case if is a top ological manifold.
is X
otal Surgery Obstruction Theorem
• T
= 0 if and for 5 only if s X Sn X n
∈ ≥
homotopy equivalent to an -dimensional
X is n
top ological manifold. 4
The pro of of the Total Surgery
Obstruction Theorem
The pro of is a translation into algebra of the
two-stage Browder-Novikov-Sullivan-Wall ob-
struction for the existence of a top ological man-
ifold in the homotopy typ e of a Poincar e com- :
plex X
The image ; of is t X Hn 1 X L s X
• ∈ − •
that = 0 if and only if the Spi-
such t X
normal bration : has a vak ν X BG
X →
top ological reduction . ν : X BTOP X →
e
= 0 then is the im- If t X s X Sn X
• ∈
of the surgery obstruction age σ f, b
∗ ∈
[ ] of the normal map : Ln Z π1 X f M
→
determined by a choice of X, b : ν ν
M → X
: . lift νX X BTOP →e
e
= 0 if and only if there exists a reduc- s X
•
: for which = 0. tion νX X BTOP σ f, b → ∗
e 5
The structure invariant
structure invariant of a homotopy equiv-
• The
: of -dimensional top o- alence h N M n
→
logical manifolds is the cob ordism class
= s h C, ψ M Sn+1
∈
a globally acyclic -dimensional quadratic
of n
Poincar e complex The stalks
C, ψ . C x
measure the failure of to have x M h
∈
acyclic p oint inverses, with
1
= H C x H h− x x
∗ ∗ →{ }
1
= H h x x M . +1 − ∗ ∈
f
acyclic p oint inverses if and only if h has
•
= 0. In particular, this is the H C x
∗
is a homeomorphism. if
case h
Invariant Theorem
• Structure
= 0 if and for 5 only s h M n Sn+1
∈ ≥
is homotopic to a homeomorphism.
if h 6
The pro of of the Structure Invariant
Theorem I
The pro of is a translation into algebra of the
two-stage Browder-Novikov-Sullivan-Wall ob-
struction for the uniqueness of top ological man-
ifold structures in a homotopy typ e :
image ; of is such the t h Hn M L s h
• ∈ •
= 0 if and only if the normal in-
that t h
variant can be trvialized
1
: h− ∗νN νM M L0 G/T OP
− ' {∗} → '
and only if 1 : extends if h M N M M
∪ ∪ → ∪
to a normal bordism
: ; [0 1]; 0 1 f, b W M,N M , ,
→ × { } { }
= 0 then is the if t h s h M Sn+1
• ∈
image of the surgery obstruction
[ ] σ f, b L π1 M . n+1 Z
∗ ∈ 7
The pro of of the Structure Invariant
Theorem I I
= 0 if and only if there exists a normal s h
•
rdism which is a simple homotopy
bo f, b
equivalence.
to work with simple -groups here, Have L
•
to take advantage of the rdism the-
s-cob o
orem.
pro of. The mapping cylinder
• Alternative : of h N M
→
[0 1] P = M N ,
∪h ×
+ 1 -dimensional geometric an
de nes n
Poincar e pair with manifold b ound- P, M N
∪
ry, such that is homotopy equivalent to
a P
The structure invariant is the rel to-
M . ∂
tal surgery obstruction
= =
s h s P P M . +1 ∂ Sn+1 Sn
∈ 8
The simply-connected case
For = 1 the algebraic surgery ex- π1 X
• { }
act sequence breaks up
0 ; 0 1 Sn X Hn 1 X L Ln Z
→ → − • → − →
total surgery obstruction The s X Sn X
• ∈
injectively to the reducibility ob-
maps TOP
struction ; of the Spi- t X Hn 1 X L
∈ − •
normal bration . Thus for 5 a vak ν n
X ≥
simply-connected geometric
n-dimensional
is homotopy equiva- oincar e complex
P X
lent to an top ological man-
n-dimensional
: admits a if and only if ifold ν X BG
X →
: . TOP reduction ν X BTOP X →
e
structure invariant The s h M Sn+1
• ∈
maps injectively to the normal invariant
; = [ ]. Thus for t h Hn M L M,G/TOP
∈ •
is homotopic to a homeomorphism n 5 h
≥
if and only if : . t h M G/T OP
' {∗} → 9
The geometric surgery exact sequence
TOP
structure set of a top olog- The S M
•
manifold is the set of equivalence
ical M
of homotopy equivalences : classes h N
→
top ological manifolds , with
M from N
there exist a homeomorphism : h h if g
∼ 0
a homotopy : . N N and hg h N M
0 → ' 0 0 →
rem Quinn, R. The geometric surgery
• Theo
= dim 5 sequence for exact n M ≥
TOP
[ ]
L π1 M M n+1 Z S
···→ →
[ ] [ ] M,G/TOP Ln Z π1 M
→ →
is isomorphic to the relevant portion of the
algebraic surgery exact sequence
[ ]
L π1 M M +1 n+1 Z Sn ···→ →
A
; [ ] Hn M L Ln Z π1 M → • −→
TOP k k 1
. =
with M D ,M S M +1 S Sn+k ∂ × × −
TOP n n
= = 0. Example S S
S Sn+1 10
The image of the assembly map
rem For any nitely presented group
• Theo
image of the assembly map
π the
1 ; [ ] A : Hn K π, L Ln Z π
• →
the subgroup of [ ] consisting of
is Ln Z π
surgery obstructions of normal the σ f, b
∗
maps : of closed -dimensional f, b N M n
→
manifolds with = .
π1 M π
are many calculations of the image
• There
for nite , notably the Oozing Con-
of A π
jecture proved by Hambleton-Milgram-Taylor-
Williams. 11
Statement of the Novikov conjecture
-genus of an -dimensional manifold The n
• L
a collection of cohomology classes
M is
4
; which are determined M H M Q
L ∈ ∗
y the Pontrjagin classes of : b ν M
M →
In general, is not a homo- BTOP . M
L
topy invariant.
Hirzebruch signature theorem for a 4 - The k
• manifold dimensional M
2k
= [ ] signature H M , M , M Z
∪ hL i∈
ws that part of the -genus is homotopy
sho L
invariant.
Novikov conjecture for a discrete group
• The
that the higher signatures for any man-
π is
ifold with =
M π1 M π
= [ ] 1 ; σx M x M , M Q x H∗ K π, Q
h ∪L i∈ ∈
are homotopy invariant. 12
Algebraic formulation of the Novikov
conjecture
rem The Novikov conjecture holds for
• Theo
group if and only if the rational assem-
a π
bly maps
: 1 ; = 1 ; A Hn K π, L Q Hn 4 K π, Q
• ⊗ − ∗
[ ] Ln Z π1 M Q
→ ⊗
are injective.
rivially true for nite . T π
•
eri ed for many in nite groups using al- V π
•
gebra, geometric group theory, ras,
C∗ -algeb
etc. See Pro ceedings of 1993 Ob erwolfach
conference LMS Lecture Notes 226,227
for state of the art in 1995, not substan-
tially out of date. 13
Statement of the Borel conjecture
-dimensional Poincar e duality group An n π
•
is a discrete group such that the classifying
space 1 is an -dimensional Poincar e
K π, n
complex.
be in nite and torsion-free. π must
•
Borel conjecture is that for every - The n
•
Poincar e duality group there
dimensional π
-dimensional manifold an aspherical
exists n
1 with M K π, '
TOP
= 0
S M .
This is top ological rigidity: every homo-
y equivalence : is conjec- top h N M
→
tured to be homotopic to a homeomor-
phism. The conjecture also predicts higher
rigidity
TOP k k 1
= 0 0 S M D ,M S − k .
∂ × × ≥ 14
Algebraic formulation of the Borel
conjecture
rem For 5 the Borel conjecture Theo n
• ≥
holds for an Poincar e group
n-dimensional
and only if 1 = 0 1 π if s K π, Sn K π,
∈
and the assembly map
: 1 ; [ ]
A H K π, L π1 M + n+k L n k Z
• →
injective for = 0 and an isomorphism
is k
for k 1.
≥
eri ed for many Poincar e duality groups
• V
, with 1 = = . π Sn K π, L0 Z Z
n
rue in the classical case = , 1 = T π Z K π,
• n
which was crucial in the extension due
T ,
to Kirby-Sieb enmann ca. 1970 of the
1960's Browder-Novikov-Sullivan-Wall surgery
cat- ry from the di erentiable and
theo PL
egories to the top ological category. 15
The 4-p erio dic algebraic surgery exact
sequence
4-p erio dic algebraic surgery exact
• The
is de ned for any space sequence X
A
; [ ] Hn X L Ln Z π1 X
···→ • −→
; Sn X Hn 1 X L ...
→ → − • →
with and the - = 0 L0 L Z G/T OP A L
×
theory assembly map. sequence
• Exact
; Hn X L0 Z Sn X Sn X ...
···→ → → →
4-p erio dic total surgery obstruction The s X
• ∈
of an -dimensional geometric Poincar e
Sn X n
is the image of . complex X s X Sn X
∈ 16
Homology manifolds I
-dimensional compact homol- Every n ANR
•
manifold is homotopy equivalent to
ogy M
nite -dimensional geometric Poincar e
a n
complex West
total surgery obstruction The s M Sn M
• ∈
an -dimensional compact homol-
of n ANR
manifold is the image of the Quinn
ogy M
resolution obstruction ; . i M Hn M L0 Z
∈
The 4-p erio dic total surgery obstruction is
= 0 . s M Sn M ∈
H
homology manifold structure set The S M
•
a compact homology manifold
of ANR M
is the set of equivalence classes of simple
y equivalences : from homotop h N M
→
manifolds , with if top ological N h h
∼ 0
there exist an rdism ; and
s-cob o W N, N0
an extension of a simple homotopy h h to
∪ 0
; [0 1]; 0 1 . equivalence W N, N N , ,
0 → × { } { } 17
Homology manifolds I I
rem Bryant-Ferry-Mio-Weinb erger
• Theo
i The 4-p erio dic total surgery obstruction
an -dimensional geometric Poincar e com-
of n
plex = 0 if and for X is s X Sn X
∈
only if is homotopy equivalent to n 6 X
≥
homology manifold. compact
a ANR
-dimensional compact For an
ii n ANR
manifold with 6 the 4- homology M n
≥
total surgery obstruction de- rel
p erio dic ∂
nes a bijection
H
; : M M h N M s h S Sn+1 → → 7→
n H n
= = , i.e. there S S L0 Sn+1 S Z
•
ho- a non-resolvable compact exists ANR
n
mology manifold homotopy equivalent
n
, with arbitrary resolution obstruction to S
n
. i L0 Z
∈ 18
The simply-connected surgery sp ectrum L
•
is ? Required prop erties What L
• •
= πn L Ln Z , L0 G/T OP .
• '
are the generalized homology groups
• What
; ? Will construct them as cob or- H X L
∗ •
dism groups of combinatorial sheaves over
quadratic Poincar e complexes over .
X of Z
so far, have only worked out
• Confession:
everything for a lo cally nite simplicial
complex using simplicial homology. In
X ,
principle, could use singular homology for
, but this would be even harder. space
any X
In any case, could use nerves of covers to
get Cech theory. 19
-mo dule category
The Z,X
simplicial complex. X =
•
-mo dule is a based f.g. free - A Z,X Z
•
with direct sum decomp osition
mo dule B
B = B σ . σ X
X∈
-mo dule morphism : is a A Z,X f B C
• →
morphism such that
Z-mo dule
f B σ C τ . ⊆ τ σ
X≥
Ranicki-Weiss A -mo dule Prop osition Z,X
•
map : is a chain equivalence chain f D E
→
and only if the -mo dule chain maps
if Z
: f σ, σ D σ E σ σ X
→ ∈
are chain equivalences. This illustrates
-lo cal nature of the -category .
the X Z,X 20
for -mo dules
Assembly Z,X
the universal covering : to Use p X X
• →
de ne the assembly functor
f
: -mo dules [ ]-mo dules ; A Z,X Z π1 X
{ }→{ }
= B B X B p σ . 7→ σ X P∈ f e
e e
order to extend to -theory need in- In A L
•
on the -category. Unfortu-
volution Z,X
nately, it do es not have one! The naive
-mo dule morphism : of a dual Z,X f B
→
not a -mo dule morphism : C is Z,X f∗
C B .
∗ → ∗
have to work with a chain duality,
• Instead,
-mo dule is a which the dual of a
in Z,X B
-mo dule chain complex . Ana-
Z,X T B
logue of Verdier duality in sheaf theory. 21
Dual cells
barycentric sub division of is the The X X
• 0
simplicial complex with one -simplex 1
n σ0 σ ...σn
r each sequence of simplexes in fo X
b b b 1 σ0 <σ < <σn .
···
dual cell of a simplex is the The σ X
• ∈
contractible sub complex
=
1 0 D σ, X σ0 σ ...σn σ σ X ,
{ | ≤ }⊆ 0
b oundary
with b b b
=
1 0 ∂D σ, X σ0 σ ...σn σ<σ D σ, X . { | }⊆
b b b
by Poincar e to prove duality.
• Intro duced
simplicial map : has acyclic A f M X
• → 0
p oint inverses if and only if
1
: f H f− D σ, X = H D σ, X σ X .
| ∗ ∗ ∼ ∗ ∈ 22
do -mo dule chain
Where Z,X
complexes come from?
or any simplicial map : the F f M X
• → 0
simplicial chain complex is a -
M Z,X
mo dule chain complex:
1 1
=
M σ f− D σ, X ,f− ∂D σ, X
simplicial co chain complex is The X
• −∗
-mo dule chain complex with:
a Z,X =
Z if r σ
= X σ r −| |
−∗ otherwise. 0
23
-mo dule chain duality
The Z,X
additive category of - The A Z,X Z,X
•
mo dules has a chain duality with dualizing
complex
X −∗
= Hom Hom
T B X ,B ,
Z Z,X −∗ Z
if = Hom B τ , Z r σ
Z −| |
= T B r σ τ σ
• P≥
if = 0 r σ 6 −| |
dual of a -mo dule chain complex the Z,X
•
a -mo dule chain complex
C is Z,X T C
with
Hom Hom
T C C, X C, 0 −∗ Z −∗
'Z Z,X 'Z Z
.
T X X
• 0 ' Z,X −∗ 24
The construction of the algebraic surgery
exact sequence
generalized - homology groups are The L
• •
-dimensional cob ordism groups of adjusted
the n
quadratic Poincar e complexes over
Z,X
; = Hn X L Ln Z,X .
•
Require adjustments to get . L0 G/T OP
'
Unadjusted ry is the 4-p erio dic ; L-theo Hn X L
•
. Adjust to kill . with
0 0 L0 L Z G/T OP L Z
' ×
assembly map from -mo dules The A Z,X
•
[ ]-mo dules induces to
Z π1 X
: [ ] A Ln Z,X Ln Z π1 X
→
relative groups = are the The Sn X πn A
•
1 -dimensional rdism groups of cob o n
−
quadratic Poincar e complexes over
C, ψ
with assembly an acyclic [ ]-
Z,X C X Z π1 X
mo dule chain complex.
f 25
Reference
raic -theory and top ological manifolds Algeb L
•
Mathematical Tracts 102, Cambridge 1992 26