[PDF] - MANIFOLDS AND DUALITY ANDREW RANICKI Classication of manifolds PDF Document

SLIDE 1

MANIFOLDS AND DUALITY

ANDREW RANICKI

Classication
f

manifolds

Uniqueness

Problem

Existence

Problem

Quadratic

algeb ra

Applications

1

SLIDE 2 Manifolds

An n-dimensional

manifold Mn is a top

logical

space which is lo cally homeomo rphic to Rn . { compact,

riented,

connected.

Classication
f

manifolds up to homeomo rphism. { F

r n

= 1: circle { F

r n

= 2: sphere, to rus, . . . , handleb

dy

. { F

r n ≥

3: in general imp

ssible.

2

SLIDE 3 The Uniqueness Problem

Is

every homotop y equivalence

f n-dimen-

sional manifolds

f

: Mn → Nn homotopic to a homeomo rphism? { F

r n

= 1, 2: Y es. { F

r n ≥

3: in general No. 3

SLIDE 4 The P

inca

r

e

conjecture

Every

homotop y equivalence f : M 3 → S 3 is homotopic to a homeomo rphism. { Stated in 1904 and still unsolved!

Theo

rem (n ≥ 5: Smale, 1960, n = 4: F reedman, 1983) Every homotop y equivalence f : Mn → Sn is homotopic to a homeomo rphism. 4

SLIDE 5 Old solution

f

the Uniqueness Problem

Surgery

theo ry w

rks

b est fo r n ≥ 5. { F rom no w

n

let n ≥ 5.

Theo

rem (Bro wder, Novik

v,

Sullivan, W all, 1970) A homotop y equivalence f : Mn → Nn is homotopic to a homeomo rphism if and

nly

if t w

bstructions

vanish.

The

2

bstructions
f

surgery theo ry: 1. In the top

logical K
theo

ry

f

vecto r bundles

ver N

. 2. In the algeb raic L

theo

ry

f

quadratic fo rms

ver

the fundamental group ring

Z

[π 1 (N )]. 5

SLIDE 6 T raditional surgery theo ry

Advantage:

{ Suitable fo r computations .

Disadvantages:

{ Inaccessible. { A complicated mix

f

top

logy

and algeb ra. { P assage from a homotop y equivalence to the

bstructions

is indirect . { Obstructions a re not indep endent . 6

SLIDE 7 W all's p rogramme

\The

theo ry

f

quadratic structures

n

chain complexes should p rovide a simple and sat- isfacto ry algeb raic version

f

the whole setup." { C.T.C.W all, Surgery

n

compact manifolds, 1970

Such

a theo ry is no w available. { Ranicki, Algeb raic L

theo

ry and top

logical

manifolds, 1992 7

SLIDE 8 Sieb enmann's theo rem

The

k ernel groups

f

a map f : M → N a re the relative homology groups

Kr

(x ) = Hr +1 (f− 1 (x ) → {x}) (x ∈ N ) .

Exact

sequence

· · · → Kr

(x ) → Hr (f− 1 (x )) → Hr ({x})

→ Kr−

1 (x ) → . . . .

K∗

(x ) = fo r a homeomo rphism f .

Theo

rem (Sieb enmann, 1972) A homotop y equivalence f : Mn → Nn with

K∗

(x ) = (x ∈ N ) is homotopic to a homeomo rphism. 8

SLIDE 9 New solution

f

the Uniqueness Problem

The

total surgery

bstruction s

(f )

f

a homotop y equivalence f : Mn → Nn is the cob

rdism

class

f

{ the sheaf

f Z
mo

dule chain complexes { with n-dimensional P

inca

r

e

dualit y {

ver N

{ with stalk homology K∗ (x ) (x ∈ N ).

Cob
rdism

and P

inca

r

e

dualit y a re algeb raic.

Theo

rem A homotop y equivalence f is homotopic to a homeomo rphism if and

nly

if s (f ) = 0. 9

SLIDE 10 P

inca

r

e

dualit y

Theo

rem (P

inca

r

e,

1895) The homology and cohomology

f

a compact

riented n-dimensional

manifold M a re isomo rphic:

Hn−r

(M ) ∼ = Hr (M ) (r = 0, 1, 2, . . . ) .

Denition

(Bro wder, 1962) An n-dimensional dualit y space X is a space with isomo rphisms:

Hn−r

(X ) ∼ = Hr (X ) (r = 0, 1, 2, . . . ) . 10

SLIDE 11 The Existence Problem

Is

an n-dimensional dualit y space X homotop y equivalent to an n-dimensional manifold? { F

r n

= 1, 2: Y es. { F

r n ≥

3: in general No. 11

SLIDE 12 Old solution

f

the Existence Problem

Theo

rem (Bro wder, Novik

v,

Sullivan, W all, 1970) An n-dimensional dualit y space X is homotop y equivalent to an n-dimensional manifold if and

nly

if 2

bstructions

vanish.

The

2

bstructions

(as fo r Uniqueness): 1. In the top

logical K
theo

ry

f

vecto r bundles

ver X

. 2. In the algeb raic L

theo

ry

f

quadratic fo rms

ver

the fundamental group ring

Z

[π 1 (X )].

Same

(dis)advantages as fo r the

ld

solution

f

the Uniqueness Problem. 12

SLIDE 13 The Theo rem

f

Galewski and Stern

The

k ernel groups Kr (x )

f

an n-dimensional dualit y space X t into the exact sequence

· · · → Kr

(x ) → Hn−r ({x}) → Hr (X, X\{x})

→ Kr−

1 (x ) → . . . .

K∗

(x ) = fo r a manifold.

Theo

rem (Galewski and Stern, 1977) A p

lyhedral

dualit y space X with

K∗

(x ) = (x ∈ X ) (a homology manifold) is homotop y equivalent to a manifold. 13

SLIDE 14 New solution

f

the Existence Problem

The

total surgery

bstruction s

(X )

f

and

n-dimensional

dualit y space X is the cob

r-

dism class

f

{ the sheaf

f Z
mo

dule chain complexes { with (n − 1)-dimensional P

inca

r

e

dualit y {

ver X

{ with stalk homology K∗ (x ) (x ∈ X ).

Cob
rdism

and P

inca

r

e

dualit y a re algeb raic.

Theo

rem A dualit y space X is homotop y equivalent to a manifold if and

nly

if s (X ) = 0. 14

SLIDE 15 Quadratic algeb ra

Chain

complexes with the homological p rop- erties

f

manifolds and dualit y spaces.

An n-dimensional

dualit y complex is a chain complex

Cn

d → Cn−

1

d → Cn−

2 → . . . → C (d 2 = 0) with isomo rphisms

Hn−r

(C ) ∼ = Hr (C ) (r = 0, 1, 2, . . . ) . { generalized quadratic fo rms

cob
rdism
f

dualit y complexes 15

SLIDE 16 Lo cal and global dualit y complexes

X

= connected space

The

global surgery group Ln (Z [π 1 (X )])

f

W all is the cob

rdism

group

f n-dimensional

dualit y complexes

f Z

[π 1 (X )]-mo dules. { Generalized Witt groups.

The

lo cal surgery group Hn (X ; L (Z )) is the cob

rdism

group

f n-dimensional

dualit y complexes

f Z
mo

dule sheaves

ver X

. { Generalized homology with co ecients

L∗

(Z ). 16

SLIDE 17 The surgery exact sequence

Theo

rem The lo cal and global surgery groups a re related b y the exact sequence

. . . → Hn

(X ; L (Z ))

A → Ln

(Z [π 1 (X )])

→ Sn

(X ) → Hn− 1 (X ; L (Z )) → . . . .

The

assembly map A is the passage from lo cal to global dualit y .

The

structure group Sn (X ) is the cob

r-

dism group

f

(n− 1)-dimensional lo cal dualit y complexes

ver X

which a re globally underlinetrivial. 17

SLIDE 18 The total surgery

bstructions
Uniqueness:

the total surgery

bstruction
f

a homotop y equivalence f : Mn → Nn

s

(f ) ∈ Sn +1 (N ) . { s (f ) is the cob

rdism

class

f

the n- dimensional globally trivial lo cal dualit y complex with stalk homology the k ernels K∗ (x ) (x ∈ N ).

Existence:

the total surgery

bstruction
f

an n-dimensional dualit y space X

s

(X ) ∈ Sn (X ) . { s (X ) is the cob

rdism

class

f

the (n − 1)-dimensional globally trivial lo cal dualit y complex with stalk homology the k ernels K∗ (x ) (x ∈ X ). 18

SLIDE 19 T

p
logy

and homotop y theo ry

The

dierence b et w een the top

logy
f

manifolds and the homotop y theo ry

f

dualit y spaces = the dierence b et w een the cob

r-

dism theo ries

f

the lo cal and global dualit y complexes. manifolds

− →

lo cal dualit y

  



 

A

dualit y spaces −

→

global dualit y

Converse
f

P

inca

r

e

dualit y : A dualit y space with sucient lo cal dualit y is homotop y equivalent to a manifold. 19

SLIDE 20 The Novik

v

and Bo rel conjectures

The

Novik

v

conjecture

n

the homotop y inva riance

f

the higher signatures is algeb raic: { A : H∗ (Bπ ; L (Z )) → L∗ (Z [π ]) is ratio- nally injective, fo r every group π .

The

Bo rel conjecture

n

the existence and uniqueness

f

aspherical manifolds is algeb raic : { A : H∗ (Bπ ; L (Z )) → L∗ (Z [π ]) is an isomo rphism if Bπ is a dualit y space.

The

va rious solution metho ds can no w b e turned into algeb ra : { top

logy

, geometry , analysis (C∗

algeb

ra), index theo rems, . . . . 20

SLIDE 21 Applications

algeb

raic computations

f

the L

groups

{ numb er theo ry

singula

r spaces { algeb raic va rieties

dierential

geometry { hyp erb

lic

geometry

non-compact

manifolds { controlled top

logy
3-

and 4-dimensional manifolds 21