[PDF] - CUTTING AND PASTING MANIFOLDS FROM THE ALGEBRAIC POINT OF VIEW PDF Document

SLIDE 1

CUTTING AND PASTING MANIFOLDS FROM THE ALGEBRAIC POINT OF VIEW

ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aa r 1

SLIDE 2 Cutting and P asting

Cut

a closed n-dimensional manifold M

M

= M 1 ∪ M 2 with M 1, M 2 manifolds with b

unda

ry

∂M

1 = ∂M 2

P

aste together M 1 and M 2 using an isomo rphism

h

: ∂M 1 → ∂M 2 to

btain

a new closed n-dimensional manifold

M′

= M 1 ∪h M 2

What

a re the inva riants

f

manifolds which do not change under cutting and pasting? 2

SLIDE 3 Schneiden und Kleb en

J
anich

(1968) cha racterized signature and Euler cha racteristic as cut and paste inva riants.

Ka

rras, Kreck, Neumann and Ossa (1973) dened SK

groups,

universal groups

f

cut and paste inva riants.

Applications

to index

f

elliptic

p

erato rs.

Some

recent applications

f

cut and paste metho ds to higher signatures, L 2

cohomology

{ Leichtnam, Lott, L uck, W einb erger, . . . 3

SLIDE 4 The b

rdism SK
groups
n

(X ) = b

rdism
f

maps from closed n- dimensional manifolds

f

: Mn → X

Denition SKn

(X ) = n (X )/∼ with

M

1 ∪g M 2 ∼ M 1 ∪h M 2 fo r any isomo rphisms g, h : ∂M 1 → ∂M 2 4

SLIDE 5 Twisted doubles

A

closed n-dimensional manifold M is a t wisted double if

M

= N ∪h N fo r n-dimensional manifold with b

unda

ry (N, ∂N ) and automo rphism h : ∂N → ∂N .

Lemma

A map f : M → X from a closed

n-dimensional

manifold M rep resents in

SKn

(X ) if and

nly

if f : M → X is b

rdant

to a t wisted double.

Pro
f

Whitehead identit y in b

rdism

M

1∪gM 2 +M 2∪hM 3 = M 1∪hgM 3 ∈ n (X ) with M 3 = M 1 . 5

SLIDE 6 Main result

The

identication

f SKn

(X ) fo r n ≥ 6 with the image

f

the assembly map in the asymmetric L

theo

ry

f Z

[π 1 (X )].

Geometric

realization

f

algeb raic result: { A symmetric P

inca

r

e

complex is an algeb raic t wisted double if and

nly

if it is null-cob

rdant

as an asymmetric P

inca

r

e

complex.

Identication

almost p roved in High dimensional knot theo ry (Sp ringer, 1998) 6

SLIDE 7 Symmetric L

theo

ry (I.)

A

= ring with involution

An n-dimensional

symmetric P

inca

r

e

complex (C, φ ) is an n-dimensional f.g. free

A-mo

dule chain complex

C

: · · · → 0 → Cn → · · · → C 1 → C with a chain equivalence

φ

: Cn−∗ = HomA (C, A)∗−n → C such that φ ≃ φ∗ , and higher symmetry conditions.

Cob
rdism
f

symmetric P

inca

r

e

complexes

Ln

(A) = cob

rdism

group (Mishchenk

,

R.) 7

SLIDE 8 Symmetric L

theo

ry (I I.)

Symmetric L
groups

= W all quadratic L

groups

mo dulo 2-p rima ry to rsion

Ln

(A) ⊗ Z [1/2] ∼ = Ln (A) ⊗ Z [1/2]

L

4∗ (Z ) = Z (signature)

The

symmetric signature

f

an n-dimensional manifold M

σ∗

(M ) = (C (

M

), φ ) ∈ Ln (Z [π 1 (M )])

Symmetric

signature map

n

b

rdism

σ∗

: ∗ (X ) → L∗ (Z [π 1 (X )]) 8

SLIDE 9 Asymmetric L

theo

ry (I.)

An n-dimensional

asymmetric P

inca

r

e

complex (C, φ ) is an n-dimensional f.g. free

A-mo

dule chain complex

C

: · · · → 0 → Cn → · · · → C with a chain equivalence

φ

: Cn−∗ = HomA (C, A)∗−n → C (no symmetry condition)

Cob
rdism
f

asymmetric P

inca

r

e

complexes

LAsyn

(A) = cob

rdism

group

F
rgetful

maps Ln (A) → LAsyn (A) 9

SLIDE 10 Asymmetric L

theo

ry (I I.)

2-p

erio dic

LAsyn

(A) ∼ = LAsyn +2 (A)

Odd-dimensional

asymmetric L

groups

van- ish

LAsy

2∗ +1 (A) =

Even-dimensional

asymmetric L

groups

a re la rge, e.g.

LAsy

(Z ) =

∞

Z ⊕

∞

Z

2 ⊕

∞

Z

4

Asymmetric

signature

f n-dimensional

manifold M

Asyσ∗

(M ) = (C (

M

), φ ) ∈ LAsyn (Z [π 1 (M )]) 10

SLIDE 11 Algeb raic t wisted doubles

Theo

rem An n-dimensional symmetric P

inca

r

e

complex (C, φ )

ver A

is an algeb raic t wisted symmetric P

inca

r

e

double if and

nly

if (C, φ ) = 0 ∈ LAsyn (A)

Pro
f

Chapter 30

f

High dimensional knot theo ry

Example

If M = N ∪h N is a t wisted double manifold then C (M ) → C (N, ∂N ) deter- mines an asymmetric P

inca

r

e

null-cob

rdism
f

the symmetric P

inca

r

e

complex

f M

, so that

σ∗

(M ) ∈ k er (Ln (Z [π 1 (M )]) → LAsyn (Z [π 1 (M )])) 11

SLIDE 12 Recognizing t wisted doubles

Theo

rem F

r n ≥

6 an n-dimensional manifold M is a t wisted double if and

nly

if

Asy

([M ]L ) = 0 ∈ LAsyn (Z [π 1 (M )])

Pro
f

The asymmetric signature is the Quinn (1979)

bstruction

to the existence

f
p

en b

k

structure

n M

M

= T (h : F → F ) ∪ ∂F × D 2

(h,

id.) = rel ∂ automo rphism

f

(n − 1)- dimensional manifold with b

unda

ry (F, ∂F ).

F
r n ≥

6

p

en b

k

if and

nly

if t wisted double. 12

SLIDE 13 Assembly

Assembly

map in symmetric L

theo

ry

A

: Hn (X ; L (Z )) → Ln (Z [π 1 (X )]) fo r any space X , with π∗ (L (Z )) = L∗ (Z ).

Every n-dimensional

manifold M has an L

theo

ry

rientation

[M ]L ∈ Hn (M ; L (Z )) with A([M ]L ) = σ∗ (M ) ∈ Ln (Z [π 1 (M )])

Symmetric

signature facto rs through assembly

σ∗

: n (X ) → Hn (X ; L (Z )) A

→ Ln

(Z [π 1 (X )])

Assembly

map in asymmetric L

theo

ry

Asy

: Hn (X ; L (Z ))

A

→ Ln

(Z [π 1 (X )])

→ LAsyn

(Z [π 1 (X )]) 13

SLIDE 14 The identication

f

the b

rdism

SK

groups
Co

rolla ry F

r

any space X and n ≥ 6 the asymmetric signature denes an isomo r- phism

SKn

(X ) ∼ = im (Asy : Hn (X ; L (Z )) → LAsyn (Z [π 1 (X )]))

Pro
f

Theo rem gives that

SKn

(X ) ∼ = im (Asy σ∗ : n (X ) → LAsyn (Z [π 1 (X )])) with

σ∗

: ∗ (X ) → H∗ (X ; L (Z )) ; (f : M → X ) → f∗ [M ]

Computation
f

homotop y t yp e

f L

(Z ) (T a ylo r and Williams, 1979) sho ws that σ∗ is

nto,

so im (Asy ) = im (Asy σ∗ ) ⊆ LAsyn (Z [π 1 (X )]) 14