[PDF] - The algebraic surgery exact sequence ANDREW RANICKI (Edinburgh) PDF Document

SLIDE 1

The algebraic surgery exact sequence

ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aa r

The

algeb raic surgery exact sequence is dened fo r any space X

· · · → Hn

(X ; L• ) A

− → Ln

(Z [π 1 (X )])

→ Sn

(X ) → Hn− 1 (X ; L• ) → . . . with A the L

theo

ry assembly map. The functo r X → S∗ (X ) is homotop y inva riant.

The

2-stage

bstructions
f

the Bro wder- Novik

v-Sullivan-W

all surgery theo ry fo r the existence and uniqueness

f

top

logical

manifold structures in a homotop y t yp e a re re- placed b y single

bstructions

in the relative groups S∗ (X )

f

the assembly map A. 1

SLIDE 2 Lo cal and global mo dules

The

assembly map A : H∗ (X ; L• ) → L∗ (Z [π 1 (X )]) is induced b y a fo rgetful functo r

A

: {(Z, X )-mo dules} → {Z [π 1 (X )]-mo dules} where the domain dep ends

n

the lo cal top

logy
f X

and the ta rget dep ends

nly
n

the fundamental group π 1 (X )

f X

, which is global.

In

terms

f

sheaf theo ry A = q !p ! with

p

:

X → X

the universal covering p rojection and q :

X → {

pt.}.

The

geometric mo del fo r the L

theo

ry assembly A is the fo rgetful functo r

{geometric

P

inca

r

e

complexes}

→ {

top

logical

manifolds.} In fact, in dimensions n ≥ 5 this functo r has the same b re as A. 2

SLIDE 3 Lo cal and global quadratic P

inca

r

e

complexes

(Global)

The L

group Ln

(Z [π 1 (X )]) is the cob

rdism

group

f n-dimensional

quadratic P

inca

r

e

complexes (C, ψ )

ver Z

[π 1 (X )].

(Lo

cal) The generalized homology group

Hn

(X ; L• ) is the cob

rdism

group

f n-

dimensional quadratic P

inca

r

e

complexes (C, ψ )

ver

(Z, X ). As in sheaf theo ry C has stalks , which a re Z

mo

dule chain complexes C (x ) (x ∈ X ).

(Lo

cal-Global) Sn (X ) is the cob

rdism

group

f

(n − 1)-dimensional quadratic P

inca

r

e

complexes (C, ψ )

ver

(Z, X ) such that the

Z

[π 1 (X )]-mo dule chain complex assembly

A(C

) is acyclic. 3

SLIDE 4 The total surgery

bstruction
The

total surgery

bstruction
f

an

n-dimensional

geometric P

inca

r

e

complex

X

is the cob

rdism

class

s

(X ) = (C, ψ ) ∈ Sn (X )

f

a Z [π 1 (X )]-acyclic (n − 1)-dimensional quadratic P

inca

r

e

complex (C, ψ )

ver

(Z, X ). The stalks C (x ) (x ∈ X ) measure the failure

f X

to have lo cal P

inca

r

e

dualit y

· · · → Hr

(C (x )) → Hn−r ({x}) → Hr (X, X\{x})

→ Hr−

1 (C (x )) → Hn−r +1 ({x}) → . . .

X

is an n-dimensional homology manifold if and

nly

if H∗ (C (x )) = 0. In pa rticula r, this is the case if X is a top

logical

manifold.

T
tal

Surgery Obstruction Theo rem

s

(X ) = 0 ∈ Sn (X ) if (and fo r n ≥ 5

nly

if )

X

is homotop y equivalent to an n-dimensional top

logical

manifold. 4

SLIDE 5 The p ro

f
f

the T

tal

Surgery Obstruction Theo rem The p ro

f

is a translation into algeb ra

f

the t w

-stage

Bro wder-Novik

v-Sullivan-W

all

b-

struction fo r the existence

f

a top

logical

manifold in the homotop y t yp e

f

a P

inca

r

e

complex X :

The

image t (X ) ∈ Hn− 1 (X ; L• )

f s

(X ) is such that t (X ) = if and

nly

if the Spi- vak no rmal b ration νX : X → BG has a top

logical

reduction

νX

: X → BTOP .

If t

(X ) = then s (X ) ∈ Sn (X ) is the image

f

the surgery

bstruction σ∗

(f, b ) ∈

Ln

(Z [π 1 (X )])

f

the no rmal map (f : M →

X, b

: νM →

νX

) determined b y a choice

f

lift

νX

: X → BTOP .

s

(X ) = if and

nly

if there exists a reduction

νX

: X → BTOP fo r which σ∗ (f, b ) = 0. 5

SLIDE 6 The structure inva riant

The

structure inva riant

f

a homotop y equivalence h : N → M

f n-dimensional

top

logical

manifolds is the cob

rdism

class

s

(h ) = (C, ψ ) ∈ Sn +1 (M )

f

a globally acyclic n-dimensional quadratic P

inca

r

e

complex (C, ψ ). The stalks C (x ) (x ∈ M ) measure the failure

f h

to have acyclic p

int

inverses, with

H∗

(C (x )) = H∗ (h− 1 (x ) → {x}) =

H∗

+1 (h− 1 (x )) (x ∈ M ) .

h

has acyclic p

int

inverses if and

nly

if

H∗

(C (x )) = 0. In pa rticula r, this is the case if h is a homeomo rphism.

Structure

Inva riant Theo rem

s

(h ) = 0 ∈ Sn +1 (M ) if (and fo r n ≥ 5

nly

if ) h is homotopic to a homeomo rphism. 6

SLIDE 7 The p ro

f
f

the Structure Inva riant Theo rem (I) The p ro

f

is a translation into algeb ra

f

the t w

-stage

Bro wder-Novik

v-Sullivan-W

all

b-

struction fo r the uniqueness

f

top

logical

manifold structures in a homotop y t yp e :

the

image t (h ) ∈ Hn (M ; L• )

f s

(h ) is such that t (h ) = if and

nly

if the no rmal inva riant can b e trvialized (h− 1 )∗νN−νM ≃ {∗} : M → L 0 ≃ G/TOP if and

nly

if 1∪h : M ∪N → M ∪M extends to a no rmal b

rdism

(f, b ) : (W ; M, N ) → M × ([0, 1]; {0}, {1})

if t

(h ) = then s (h ) ∈ Sn +1 (M ) is the image

f

the surgery

bstruction

σ∗

(f, b ) ∈ Ln +1 (Z [π 1 (M )]) . 7

SLIDE 8 The p ro

f
f

the Structure Inva riant Theo rem (I I)

s

(h ) = if and

nly

if there exists a no rmal b

rdism

(f, b ) which is a simple homotop y equivalence.

Have

to w

rk

with simple L

groups

here, to tak e advantage

f

the s

cob
rdism

the-

rem.
Alternative

p ro

f.

The mapping cylinder

f h

: N → M

P

= M ∪h N × [0, 1] denes an (n + 1)-dimensional geometric P

inca

r

e

pair (P, M∪N ) with manifold b

und-

a ry , such that P is homotop y equivalent to

M

. The structure inva riant is the rel ∂ total surgery

bstruction

s

(h ) = s∂ (P ) ∈ Sn +1 (P ) = Sn +1 (M ) . 8

SLIDE 9 The simply-connected case

F
r π

1 (X ) = {1} the algeb raic surgery exact sequence b reaks up 0 → Sn (X ) → Hn− 1 (X ; L• ) → Ln− 1 (Z ) →

The

total surgery

bstruction s

(X ) ∈ Sn (X ) maps injectively to the TOP reducibilit y

b-

struction t (X ) ∈ Hn− 1 (X ; L• )

f

the Spi- vak no rmal b ration νX . Thus fo r n ≥ 5 a simply-connected n-dimensional geometric P

inca

r

e

complex X is homotop y equivalent to an n-dimensional top

logical

manifold if and

nly

if νX : X → BG admits a

TOP

reduction

νX

: X → BTOP .

The

structure inva riant s (h ) ∈ Sn +1 (M ) maps injectively to the no rmal inva riant

t

(h ) ∈ Hn (M ; L• ) = [M, G/TOP ]. Thus fo r

n ≥

5 h is homotopic to a homeomo rphism if and

nly

if t (h ) ≃ {∗} : M → G/TOP . 9

SLIDE 10 The geometric surgery exact sequence

The

structure set STOP (M )

f

a top

log-

ical manifold M is the set

f

equivalence classes

f

homotop y equivalences h : N →

M

from top

logical

manifolds N , with

h ∼ h′

if there exist a homeomo rphism g :

N′ → N

and a homotop y hg ≃ h′ : N′ → M .

Theo

rem (Quinn, R.) The geometric surgery exact sequence fo r n = dim (M ) ≥ 5

· · · → Ln

+1 (Z [π 1 (M )]) → STOP (M )

→

[M, G/TOP ] → Ln (Z [π 1 (M )]) is isomo rphic to the relevant p

rtion
f

the algeb raic surgery exact sequence

· · · → Ln

+1 (Z [π 1 (M )]) → Sn +1 (M )

→ Hn

(M ; L• ) A

− → Ln

(Z [π 1 (M )]) with STOP

∂

(M×Dk, M×Sk− 1 ) = Sn +k +1 (M ). Example STOP (Sn ) = Sn +1 (Sn ) = 0. 10

SLIDE 11 The image

f

the assembly map

Theo

rem F

r

any nitely p resented group

π

the image

f

the assembly map

A

: Hn (K (π, 1); L• ) → Ln (Z [π ]) is the subgroup

f Ln

(Z [π ]) consisting

f

the surgery

bstructions σ∗

(f, b )

f

no rmal maps (f, b ) : N → M

f

closed n-dimensional manifolds with π 1 (M ) = π .

There

a re many calculations

f

the image

f A

fo r nite π , notably the Oozing Con- jecture p roved b y Hambleton-Milgram-T a ylo r- Williams. 11

SLIDE 12 Statement

f

the Novik

v

conjecture

The L
genus
f

an n-dimensional manifold

M

is a collection

f

cohomology classes

L

(M ) ∈ H 4∗ (M ; Q ) which a re determined b y the P

ntrjagin

classes

f νM

: M →

BTOP

. In general, L (M ) is not a homotop y inva riant.

The

Hirzeb ruch signature theo rem fo r a 4k

dimensional

manifold M signature (H 2k (M ), ∪ ) = L (M ), [M ] ∈ Z sho ws that pa rt

f

the L

genus

is homotop y inva riant.

The

Novik

v

conjecture fo r a discrete group

π

is that the higher signatures fo r any manifold M with π 1 (M ) = π

σx

(M ) = x∪L (M ), [M ] ∈ Q (x ∈ H∗ (K (π, 1); Q )) a re homotop y inva riant. 12

SLIDE 13 Algeb raic fo rmulation

f

the Novik

v

conjecture

Theo

rem The Novik

v

conjecture holds fo r a group π if and

nly

if the rational assembly maps

A

: Hn (K (π, 1); L• ) ⊗ Q = Hn− 4∗ (K (π, 1); Q )

→ Ln

(Z [π 1 (M )]) ⊗ Q a re injective.

T

rivially true fo r nite π .

V

eried fo r many innite groups π using algeb ra, geometric group theo ry , C∗

algeb

ras, etc. See Pro ceedings

f

1993 Ob erw

lfach

conference (LMS Lecture Notes 226,227) fo r state

f

the a rt in 1995, not substan- tially

ut
f

date. 13

SLIDE 14 Statement

f

the Bo rel conjecture

An n-dimensional

P

inca

r

e

dualit y group π is a discrete group such that the classifying space K (π, 1) is an n-dimensional P

inca

r

e

complex.

π

must b e innite and to rsion-free.

The

Bo rel conjecture is that fo r every n- dimensional P

inca

r

e

dualit y group π there exists an aspherical n-dimensional manifold

M ≃ K

(π, 1) with

STOP

(M ) = 0 . This is top

logical

rigidit y: every homotop y equivalence h : N → M is (conjec- tured) to b e homotopic to a homeomo r- phism. The conjecture also p redicts higher rigidit y

STOP

∂

(M × Dk, M × Sk− 1 ) = (k ≥ 0) . 14

SLIDE 15 Algeb raic fo rmulation

f

the Bo rel conjecture

Theo

rem F

r n ≥

5 the Bo rel conjecture holds fo r an n-dimensional P

inca

r

e

group

π

if and

nly

if s (K (π, 1)) = 0 ∈ Sn (K (π, 1)) and the assembly map

A

: Hn +k (K (π, 1); L• ) → Ln +k (Z [π 1 (M )]) is injective fo r k = and an isomo rphism fo r k ≥ 1.

V

eried fo r many P

inca

r

e

dualit y groups

π

, with Sn (K (π, 1)) = L (Z ) = Z .

T

rue in the classical case π = Zn , K (π, 1) =

T n

, which w as crucial in the extension due to Kirb y-Sieb enmann (ca. 1970)

f

the 1960's Bro wder-Novik

v-Sullivan-W

all surgery theo ry from the dierentiable and PL catego ries to the top

logical

catego ry . 15

SLIDE 16 The 4-p erio dic algeb raic surgery exact sequence

The

4-p erio dic algeb raic surgery exact sequence is dened fo r any space X

· · · → Hn

(X ; L• ) A

− → Ln

(Z [π 1 (X )])

→ Sn

(X ) → Hn− 1 (X ; L• ) → . . . with L = L (Z ) × G/TOP and A the L

theo

ry assembly map.

Exact

sequence

· · · → Hn

(X ; L (Z )) → Sn (X ) → Sn (X ) → . . .

The

4-p erio dic total surgery

bstruction s

(X ) ∈

Sn

(X )

f

an n-dimensional geometric P

inca

r

e

complex X is the image

f s

(X ) ∈ Sn (X ). 16

SLIDE 17 Homology manifolds (I)

Every n-dimensional

compact ANR homol-

gy

manifold M is homotop y equivalent to a nite n-dimensional geometric P

inca

r

e

complex (W est)

The

total surgery

bstruction s

(M ) ∈ Sn (M )

f

an n-dimensional compact ANR homol-

gy

manifold M is the image

f

the Quinn resolution

bstruction i(M

) ∈ Hn (M ; L (Z )). The 4-p erio dic total surgery

bstruction

is

s

(M ) = 0 ∈ Sn (M ).

The

homology manifold structure set SH (M )

f

a compact ANR homology manifold M is the set

f

equivalence classes

f

simple homotop y equivalences h : N → M from top

logical

manifolds N , with h ∼ h′ if there exist an s

cob
rdism

(W ; N, N′ ) and an extension

f h∪h′

to a simple homotop y equivalence (W ; N, N′ ) → N×([0, 1]; {0}, {1}). 17

SLIDE 18 Homology manifolds (I I)

Theo

rem (Bry ant-F erry-Mio-W einb erger) (i) The 4-p erio dic total surgery

bstruction
f

an n-dimensional geometric P

inca

r

e

complex X is s (X ) = 0 ∈ Sn (X ) if (and fo r

n ≥

6

nly

if ) X is homotop y equivalent to a compact ANR homology manifold. (ii) F

r

an n-dimensional compact ANR homology manifold M with n ≥ 6 the 4- p erio dic rel ∂ total surgery

bstruction

denes a bijection

SH

(M ) → Sn +1 (M ) ; (h : N → M ) → s (h )

Sn

+1 (Sn ) = SH (Sn ) = L (Z ), i.e. there exists a non-resolvable compact ANR homology manifold n homotop y equivalent to Sn , with a rbitra ry resolution

bstruction

i(n

) ∈ L (Z ). 18

SLIDE 19 The simply-connected surgery sp ectrum

L•

What

is L• ? Required p rop erties

πn

(L• ) = Ln (Z ) , L 0 ≃ G/TOP .

What

a re the generalized homology groups

H∗

(X ; L• )? Will construct them as cob

r-

dism groups

f

combinato rial sheaves

ver

X

f

quadratic P

inca

r

e

complexes

ver Z

.

Confession:

so fa r, have

nly

w

rk

ed

ut

everything fo r a (lo cally nite) simplicial complex X , using simplicial homology . In p rinciple, could use singula r homology fo r any space X , but this w

uld

b e even ha rder. In any case, could use nerves

f

covers to get

Cech

theo ry . 19

SLIDE 20 The (Z, X )-mo dule catego ry

X

= simplicial complex.

A

(Z, X )-mo dule is a based f.g. free Z

mo

dule B with direct sum decomp

sition

B

=

σ∈X

B

(σ ) .

A

(Z, X )-mo dule mo rphism f : B → C is a

Z

mo

dule mo rphism such that

f

(B (σ )) ⊆

τ≥σ

C

(τ ) .

Prop
sition

(Ranicki-W eiss) A (Z, X )-mo dule chain map f : D → E is a chain equivalence if and

nly

if the Z

mo

dule chain maps

f

(σ, σ ) : D (σ ) → E (σ ) (σ ∈ X ) a re chain equivalences. (This illustrates the X

lo

cal nature

f

the (Z, X )-catego ry). 20

SLIDE 21 Assembly fo r (Z, X )-mo dules

Use

the universal covering p :

X → X

to dene the assembly functo r

A

: {(Z, X )-mo dules} → {Z [π 1 (X )]-mo dules} ;

B → B

(

X

) =

σ∈

X

B

(p (

σ

)) .

In
rder

to extend A to L

theo

ry need in- volution

n

the (Z, X )-catego ry . Unfo rtu- nately , it do es not have

ne!

The naive dual

f

a (Z, X )-mo dule mo rphism f : B →

C

is not a (Z, X )-mo dule mo rphism f∗ :

C∗ → B∗

.

Instead,

have to w

rk

with a chain dualit y , in which the dual

f

a (Z, X )-mo dule B is a (Z, X )-mo dule chain complex T (B ). Ana- logue

f

V erdier dualit y in sheaf theo ry . 21

SLIDE 22 Dual cells

The

ba rycentric sub division X′

f X

is the simplicial complex with

ne n-simplex

σ σ

1 . . .

σn

fo r each sequence

f

simplexes in X

σ

0 < σ 1 < · · · < σn .

The

dual cell

f

a simplex σ ∈ X is the contractible sub complex

D

(σ, X ) = {

σ σ

1 . . .

σn | σ ≤ σ

0} ⊆ X′ , with b

unda

ry

∂D

(σ, X ) = {

σ σ

1 . . .

σn | σ < σ

0} ⊆ D (σ, X ) .

Intro

duced b y P

inca

r

e

to p rove dualit y .

A

simplicial map f : M → X′ has acyclic p

int

inverses if and

nly

if (f|)∗ : H∗ (f− 1D (σ, X )) ∼ = H∗ (D (σ, X )) (σ ∈ X ) . 22

SLIDE 23 Where do (Z, X )-mo dule chain complexes come from?

F
r

any simplicial map f : M → X′ the simplicial chain complex (M ) is a (Z, X )- mo dule chain complex: (M )(σ ) = (f− 1D (σ, X ), f− 1∂D (σ, X ))

The

simplicial co chain complex (X )−∗ is a (Z, X )-mo dule chain complex with: (X )−∗ (σ )r =

  

Z

if r = −|σ|

therwise.

23

SLIDE 24 The (Z, X )-mo dule chain dualit y

The

additive catego ry A (Z, X )

f

(Z, X )- mo dules has a chain dualit y with dualizing complex (X )−∗

T

(B ) = HomZ (Hom (Z,X ) ((X )−∗, B ), Z )

T

(B )r (σ ) =

    

τ≥σ

HomZ (B (τ ), Z ) if r = −|σ| if r = −|σ|

the

dual

f

a (Z, X )-mo dule chain complex

C

is a (Z, X )-mo dule chain complex T (C ) with

T

(C ) ≃Z Hom (Z,X ) (C, (X′ ))−∗ ≃Z HomZ (C, Z )−∗

T

((X′ )) ≃ (Z,X ) (X )−∗ . 24

SLIDE 25 The construction

f

the algeb raic surgery exact sequence

The

generalized L•

homology

groups a re the cob

rdism

groups

f

adjusted n-dimensional quadratic P

inca

r

e

complexes

ver

(Z, X )

Hn

(X ; L• ) = Ln (Z, X ) . Require adjustments to get L 0 ≃ G/TOP . Unadjusted L

theo

ry is the 4-p erio dic Hn (X ; L• ) with L 0 ≃ L (Z )×G/TOP . Adjust to kill L (Z ).

The

assembly map A from (Z, X )-mo dules to Z [π 1 (X )]-mo dules induces

A

: Ln (Z, X ) → Ln (Z [π 1 (X )])

The

relative groups Sn (X ) = πn (A) a re the cob

rdism

groups

f

(n − 1)-dimensional quadratic P

inca

r

e

complexes (C, ψ )

ver

(Z, X ) with assembly C (

X

) an acyclic Z [π 1 (X )]- mo dule chain complex. 25

SLIDE 26 Reference

Algeb

raic L

theo

ry and top

logical

manifolds Mathematical T racts 102, Camb ridge (1992) 26