[PDF] - Foundations of algebraic surgery ANDREW RANICKI (Edinburgh) PDF Document

SLIDE 1

Foundations of algebraic surgery

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

An n-dimensional

manifold M determines an n-dimensional cellula r f.g. free ab elian group chain complex

C

(M ) : Cn (M ) d

− → Cn−

1 (M ) d

− → . . . d − → C

(M ) with a P

inca

r

e

dualit y chain equivalence

C

(M )n−∗ ≃ C (M ) .

The

homology eect

f

a geometric surgery

n

a manifold M is given b y an algeb raic surgery

n

the chain complex C (M ). 1

SLIDE 2 The algeb raic surgery machine

The

algeb raic pa rt

f

the machine stud- ies chain complexes with P

inca

r

e

dualit y , which a re quadratic fo rms

n

chain complexes. Algeb raic surgery

n

such

bjects

mo dels the surgery

f

manifolds and no r- mal maps.

The

geometric pa rt reduces top

logical

surgery p roblems to algeb raic

nes.
F
r n-dimensional

no rmal maps with n ≥ 5 there is a

ne-one

co rresp

ndence

b et w een algeb raic and geometric surgery b elo w and in the middle dimensions. 2

SLIDE 3 Symmetric and quadratic P

inca

r

e

dualit y

General

theo ry applies to chain complexes

C

ver

any ring with involution A. There a re t w

t

yp es

f

P

inca

r

e

dualit y Cn−∗ ≃ C , co rresp

nding

to t yp e I and t yp e I I symmetric fo rms.

Use

symmetric P

inca

r

e

dualit y fo r surgery

n

manifolds. In general, cannot realize symmetric P

inca

r

e

surgeries

n

manifolds.

Use

quadratic P

inca

r

e

dualit y fo r surgery

n

no rmal maps.

Theo

rem The W all surgery

bstruction

group

Ln

(Z [π ]) is the cob

rdism

group

f n-dimensional

quadratic P

inca

r

e

complexes

ver Z

[π ]. 3

SLIDE 4 Geometric surgery

The n-dimensional

manifold

btained

from an n-dimensional manifold M b y surgery

n

Si × Dn−i ⊂ M

is

M′

= (M\Si × Dn−i ) ∪ Di+1 × Sn−i− 1 . Call this the eect

f

the surgery .

Can
btain M′

from M b y surgery

n

Di+1 × Sn−i−

1 ⊂ M′ .

Example

View Sn = ∂ (Di+1 × Dn−i ) as

Sn

= Si × Dn−i ∪ Di+1 × Sn−i− 1 . Surgery

n Si × Dn−i ⊂ Sn

gives

Di+1×Sn−i−

1∪Di+1×Sn−i− 1 = Si+1×Sn−i− 1 . 4

SLIDE 5 Cob

rdism

eect

f

surgery

The

trace

f

surgery

n Si × Dn−i ⊂ M

with eect M′ is the cob

rdism

(W ; M, M′ )

btained

from M × [0, 1] b y attaching an (i + 1)-handle at Si × Dn−i × {1} ⊂ M × {1}

W

= M × [0, 1] ∪ Di+1 × Dn−i .

Theo

rem (Thom, Milno r) Every (n+1)-dimensional cob

rdism

(L ; M, N ) is a union

f

the traces

f

surgeries. Pro

f

A Mo rse function

f

: (L ; M, N ) → ([0, 1]; {0}, {1}) determines a handle decomp

sition
f L
n M

L

= M × [0, 1] ∪

n

+1

k

=0

Dk × Dn

+1−k with

ne k
handle

fo r each critical p

int
f

in- dex k . 5

SLIDE 6 Homotop y eect

f

surgery

The

mapping cone

f

a map x : Si → M is the adjunction space M ∪x Di+1

btained

from M b y attaching an (i + 1)-cell.

Prop
sition

If (W ; M, M′ ) is the trace

f

a surgery

n Si ×Dn−i ⊂ M

there a re dened homotop y equivalences

W ≃ M ∪x Di+1 ≃ M′ ∪x′ Dn−i

with x : Si = Si × {0} ⊂ Si × Dn−i ⊂ M .

Surgery
n

an n-dimensional manifold at- taches an (i + 1)-cell and then detaches an (n − i)-cell.

Example

The trace (W ; Sn, Si+1 × Sn−i− 1 )

f

the surgery

n Si × Dn−i ⊂ Sn

has

W ≃ Sn ∨ Si+1 .

6

SLIDE 7 Homology eect

f

attaching a cell

The

algeb raic mapping cone

f

a chain map

f

: C → D is the chain complex C (f ) with

C

(f )r = Cr− 1⊕Dr , dC (f ) =

dC

±f dD

.
F
r i ≥

dene the chain complex

SiZ

:

· · · →

0 → Z → 0 → . . . concentrated in dimension i.

The

homology eect

f

attaching an (i + 1)-cell to M at x : Si → M is to attach an algeb raic (i + 1)-cell to C (M ) : if W = M ∪x Di+1 then

C

(W ) = C (x : SiZ → C (M ))

In

pa rticula r, Hi (W ) = Hi (M )/x is

b-

tained from Hi (M ) b y killing x ∈ Hi (M ). 7

SLIDE 8 Homology eect

f

surgery

If

(W ; M, M′ ) is the trace

f

a surgery

n

Si ×Dn−i ⊂ M

then there a re dened chain equivalences

C

(W ) ≃ C (x : SiZ → C (M ))

C

(W ) ≃ C (x′ : Sn−i− 1Z → C (M′ )) .

C

(M′ )

btained

from C (M ) b y an algeb raic surgery which kills x ∈ Hi (M ), b y rst attaching an algeb raic (i + 1)-cell and then detaching an algeb raic (n − i)-cell.

Need

P

inca

r

e

dualit y to describ e the re- lationship b et w een x ∈ Hi (M ) and x′ ∈

Hn−i−

1 (M′ ). 8

SLIDE 9 P

inca

r

e

dualit y

Cap

p ro duct with fundamental class [M ] ∈

Hn

(M ) is a chain equivalence [M ] ∩ − : C (M )n−∗ → C (M ) inducing the P

inca

r

e

dualit y isomo rphisms

Hn−∗

(M ) ∼ = H∗ (M ) .

P
inca

r

e-Lefschetz

dualit y fo r any cob

r-

dism (W ; M, M′ )

Hn

+1−∗ (W, M ) ∼ = H∗ (W, M′ ) .

Can

use P

inca

r

e

dualit y to decide which elements in Hi (M ) can b e rep resented b y

Si × Dn−i ⊂ M

and so killed b y surgery . Rega rding dualit y as a quadratic fo rm, can

nly

kill isotropic elements. 9

SLIDE 10 Principle

f

Algeb raic Surgery

F
r

any cob

rdism
f n-dimensional

manifolds (W ; M, M′ ) the chain homotop y t yp e

f C

(M′ ) and its P

inca

r

e

dualit y can b e

btained

from { the chain homotop y t yp e

f C

(M ) and its P

inca

r

e

dualit y { the chain homotop y class

f

the chain map j : C (M ) → C (W, M′ ) { j [M ] = 0 ∈ Hn (W, M′ )

n

the chain level using algeb raic surgery

n

symmetric P

inca

r

e

complexes.

An

algeb raic surgery co rresp

nds

to a se- quence

f

geometric surgeries. 10

SLIDE 11 Symmetric P

inca

r

e

complexes

An n-dimensional

symmetric P

inca

r

e

complex (C, φ ) is an n-dimensional f.g. free chain complex C with mo rphisms

φs

: Cr = HomZ (Cr, Z ) → Cn−r +s (s ≥ 0) such that (up to signs)

dφs

+φsd∗ +φs− 1 +φ∗

s−

1 = : Cr → Cn−r +s− 1 with s ≥ 0, φ− 1 = and

φ

: Cn−∗ = HomZ (C, Z )n−∗ → C is a chain equivalence.

Symmetric

fo rm

n

chain complex.

Theo

rem (Mishchenk

)

An n-dimensional manifold M determines an n-dimensional symmetric P

inca

r

e

complex (C (M ), φ ), with

φ

= [M ] ∩ − : C (M )n−∗

≃

− → C

(M ) 11

SLIDE 12 Symmetric algeb raic surgery

An

algeb raic surgery

n

(C, φ ) has input a chain map j : C → D with chain homotop y

δφ

: jφ 0j∗ ≃ : Dn−∗ → D . The eect is the n-dimensional symmetric P

inca

r

e

complex (C′, φ′ ) with

C′

r

= Cr ⊕ Dr +1 ⊕ Dn−r +1 ,

dC′

=

  

dC ±φ

0j∗

±j dD δφ

(dD )∗

  

Generalization
f

the

p

eration which re- places a symmetric fo rm (K, λ = λ∗ : K →

K∗

) fo r any x ∈ K with λ (x )(x ) = (isotropic) b y the sub quotient fo rm (K′, λ′ ) = ({y ∈ K | λ (x )(y ) = 0}/x, [λ ]) 12

SLIDE 13 Algeb raic and geometric surgery

Surgery
n Si × Dn−i ⊂ M

determines algeb raic surgery

n

(C (M ), φ ) with input

j

: C (M ) → Sn−iZ a co cycle rep resenting the P

inca

r

e

dual j ∈ Hn−i (M )

f x

= [Si ] ∈ Hi (M ), and δφ determined b y fram- ing

f Si ⊂ M

.

Theo

rem The symmetric P

inca

r

e

complex (C (M′ ), φ′ )

f

the geometric eect M′ is the eect

f

the algeb raic surgery

n

(C (M ), φ ).

Exercise

W

rk
ut

the algeb raic surgery co rresp

nding

to the geometric surgery

n

Si × Dn−i ⊂ Sn

with eect Si+1 × Sn−i− 1 . 13

SLIDE 14 The algeb raic eect

f

a surgery

The

Theo rem is an example

f

the Alge- b raic Surgery Principle in action.

If

(W ; M, M′ ) is the trace

f

surgery

n Si×

Dn−i ⊂ M

then

j

: C (M ) → C (W, M′ ) ≃ Sn−iZ .

W

rite C (M ) = C , and let x ∈ Ci b e cycle b eing killed, y = j ∈ Cn−i the dual co cycle.

The

chain complex C (M′ ) is chain equivalent to

C′

: · · · → Cn−i

d⊕y

− − → Cn−i−

1 ⊕ Z d⊕

− − − → Cn−i−

2

→ · · · → Ci+2

d⊕

− − − → Ci+1 ⊕ Z d⊕x − − − → Ci → . . .

14

SLIDE 15 Comp

sition
f

surgeries

Supp
se

given a surgery

n Si × Dn−i ⊂ M

with eect M′ and trace (W ; M, M′ ), and then a surgery

n Sk × Dn−k ⊂ M′

with ef- fect M′′ and trace (W ′ ; M′, M′′ ). Can apply the Principle to the union cob

rdism

(W ′′ ; M, M′′ ) = (W ; M, M′ ) ∪ (W ′ ; M′, M′′ ) to recover C (M′′ ) from C (M ) and C (W ′′, M′′ ).

If k

= i + 1 then C (W ′′, M′′ ) is chain equivalent to

· · · →

0 → Z λ

− → Z →

0 → . . . concentrated in dimensions n − i, n − i + 1, with λ ∈ Z the algeb raic intersection num- b er

f

the co co re {0} × Sn−i− 1 ⊂ M′ and the co re Si+1 × {0} ⊂ M′

f

the surgeries. 15

SLIDE 16 Handle cancellation

Surgery
n

Si × Dn−i ⊂ Si × Dn−i ∪Di+1 × Sn−i−

1 = Sn has trace (W ; Sn, Si+1 × Sn−i− 1 ) a punctured Si+1 × Dn−i . Reverse the roles

f

i

+ 1, n − i − 1. Surgery

n

Si+1 × Dn−i−

1 ⊂ Si+1 × Sn−i− 1 has trace (W ′ ; Si+1 × Sn−i− 1, Sn ) a punctured Di+2 × Sn−i− 1 .

The

union W ∪ W ′ is a double punctured

Sn

+1 = Si+1 × Dn−i ∪ Di+2 × Sn−i− 1

In

this case λ = 1, and there is an isomo r- phism (W ∪ W ′ ; Sn, Sn ) ∼ = Sn × ([0, 1]; {0}, {1}) The t w

surgeries

have cancelled each

ther
ut.

Dra w the picture fo r i = 0, n = 1 ! 16

SLIDE 17 Surgery and the

rthogonal

group

Can

va ry extension

f x

: Si ⊂ M to Si ×

Dn−i ⊂ M

b y maps ω : Si → O (n − i). Co r- resp

nds

to dierent δφ 's. Given ω : Si →

O

(n − i) dene emb edding

eω

: Si × Dn−i → Sn = Si × Dn−i ∪ Di+1 × Sn−i− 1 ; (x, y ) → (x, ω (x )(y )) .

Surgery
n Sn

killing eω gives sphere bundle

S

(ω )

f

the (n−i)-plane vecto r bundle

ver

Si+1

with clutching map ω

ω ∈ πi

(O (n − i)) = πi+1 (BO (n − i)) ,

Sn−i−

1 → S (ω ) → Si+1 ,

S

(ω ) = Di+1 × Sn−i− 1 ∪ω Di+1 × Sn−i− 1 . The trace

f

the surgery is cl (E (ω )\Dn +1 ), with Dn−i → E (ω ) → Si+1 the disk bundle.

Exercise

W

rk
ut

the algeb raic eect! 17

SLIDE 18 Quadratic P

inca

r

e

complexes

An n-dimensional

quadratic P

inca

r

e

complex (C, ψ ) is an n-dimensional f.g. free chain complex C with mo rphisms

ψs

: Cr → Cn−r−s (s ≥ 0) such that (up to signs)

dψs

+ψsd∗ +ψs+1 +ψ∗

s+1

= : Cr → Cn−r−s− 1 and

ψ

+ (ψ )∗ : Cn−∗ → C a chain equivalence.

Quadratic

fo rm

n

chain complex.

Can

dene algeb raic surgery as in the symmetric case. 18

SLIDE 19 Cob

rdism
f

quadratic P

inca

r

e

complexes

The n-dimensional

quadratic P

inca

r

e

complexes (C, ψ ), (C′, ψ′ ) a re cob

rdant

if there exists an algeb raic surgery

n

(C ⊕ C′, ψ ⊕ −ψ′ ) with eect (C′′, ψ′′ ) such that H∗ (C′′ ) = 0. (This is equivalent to the P

inca

r

e-Lefschetz

dualit y denition).

Theo

rem The W all surgery

bstruction

group

Ln

(A)

f

a ring with involution A is isomo rphic to the group

f

cob

rdism

classes

f n-dimensional

quadratic P

inca

r

e

complexes

ver A.
L

2i (A) is the Witt group

f

nonsingula r (−)i

quadratic

fo rms

ver A.

Such fo rms a re p recisely the 2i-dimensional quadratic P

inca

r

e

complexes (C, ψ )

ver A

with Cr = fo r r = i. Simila rly fo r L 2i+1 (A) and fo r- mations, with Cr = fo r r = i, i + 1. 19

SLIDE 20 The algeb raic surgery

bstruction
Theo

rem An n-dimensional no rmal map (f, b ) :

M → X

determines an n-dimensional quadratic P

inca

r

e

complex (C, ψ )

ver Z

[π 1 (X )] with homology the k ernel mo dules

H∗

(C ) = K∗ (M ) = k er(

f∗

: H∗ (

M

) → H∗ (

X

)) . The surgery

bstruction
f

(f, b ) is the cob

r-

dism class

f

(C, ψ )

σ∗

(f, b ) = (C, ψ ) ∈ Ln (Z [π 1 (X )]) . Thus (C, ψ ) = if (and fo r n ≥ 5

nly

if ) (f, b ) is no rmal b

rdant

to a homotop y equivalence.

Half

the p ro

f

is b y the no rmal map ver- sion

f

the Principle

f

Algeb raic Surgery: a surgery

n

(f, b ) determines a quadratic P

inca

r

e

surgery

n

the k ernel complex (C, ψ ). The

ther

half is b y the geometric realiza- tion fo r n ≥ 5

f

quadratic surgeries b elo w and in the middle dimension. 20