[PDF] - ALGEBRAIC TRANSVERSALITY ANDREW RANICKI (Edinburgh) PDF Document

SLIDE 1

ALGEBRAIC TRANSVERSALITY

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

Ho

w do es

ne

build (n + 1)-dimensional manifolds from n-dimensional manifolds? { Fib re bundles

ver S

1 . { Op en b

ks.
Ho

w do es

ne

nd n-dimensional submani- folds inside (n + 1)-dimensional manifolds? { Geometric transversalit y .

What

is algeb raic transversalit y? 1

SLIDE 2 Time scale

2-

and 3-dimensional manifolds : 1900 {

Algeb

raic va rieties : 1920 {

Algeb

raic K

and L
theo

ry : 1940 {

High-dimensional

manifolds : 1960 {

3-

and 4-dimensional manifolds, TQFT : 1980 { 2

SLIDE 3 Aims

1

: Give homological criterion

n

a high- dimensional manifold M which is necessa ry and sucient to decomp

se M

as a b re bundle

ver S

1 . { Algeb raic K

theo

ry

f

chain complexes.

2

: Lik ewise fo r

p

en b

k

decomp

sition.

{ Algeb raic L

theo

ry

f

chain complexes with P

inca

r

e

dualit y . 3

SLIDE 4 Fib re bundles

ver S

1

The

mapping to rus

f

a map h : F → F is the identication space

T

(h ) = (F × [0, 1])/ ∼ with (x, 0) ∼ (h (x ), 1).

If F

is a closed n-dimensional manifold and

h

is an automo rphism then T (h ) is a closed (n + 1)-dimensional manifold which is a b re bundle

ver S

1 , with p rojection

T

(h ) → [0, 1]/(0 ∼ 1) = S 1 ; [x, t ] → [t ] . 4

SLIDE 5 Brief histo ry

f

b re bundles

ver S

1

Stallings

(1961) : a sucient group- and homotop y-theo retic criterion fo r a 3-dimensional manifold M to b re

ver S

1 .

Bro

wder and Levine (1964) (π 1 (M ) = Z ) and F a rrell (1970) (any π 1 (M )) : necessa ry and sucient conditions fo r an n-dimensional manifold M to b re

ver S

1 , fo r n ≥ 6 : { a nitely dominated innite cyclic cover M , { the vanishing

f

the Whitehead to rsion

τ

= τ (M → T (ζ )) ∈ Wh (π 1 (M )) with ζ : M → M generating covering translation.

Novik
v,

F a rb er and P azhitnov (1981{ ) :

S

1

valued

Mo rse theo ry . 5

SLIDE 6 F redholm lo calization

A

= ring.

A[z, z−

1 ] = Laurent p

lynomial

extension.

Denition

: A squa re matrix ω in A[z, z− 1 ] is F redholm if cok er (ω ) is a f.g. p rojective

A-mo

dule.

Example

: ω = 1 − z is F redholm.

Denition

:

=

F redholm matrices in A[z, z− 1 ]. − 1A[z, z− 1 ] = the noncommutative lo calization

f A[z, z−

1 ] inverting each ω ∈ .

Example

: if A = K is a eld then − 1A[z, z− 1 ] = K (z ) is the function eld.

K

1 (− 1A[z, z− 1 ]) = K 1 (A[z, z− 1 ])⊕Aut (A). 6

SLIDE 7 Recognizing b re bundles homologically

M

= compact n-dimensional manifold with innite cyclic cover M , such that

π

1 (M ) = π × Z , π 1 (M ) = π .

A

= Z [π ] , A[z, z− 1 ] = Z [π × Z ] .

Theo

rem 1 M is nitely dominated if and

nly

if H∗ (M ; − 1A[z, z− 1 ]) = 0.

Theo

rem 2 If n ≥ 6 then M is a b re bundle

ver S

1 if and

nly

if M is nitely dominated with − 1A[z, z− 1 ]-co ecient Reidemeister-Whitehead to rsion (= F a rrell

bstruction)

is 0, that is

τ

(M ; − 1A[z, z− 1 ]) =

∈ K

1 (− 1A[z, z− 1 ])/({±(π × Z )} ⊕ Aut (A)) = Wh (π × Z ) . 7

SLIDE 8 Op en b

ks
The

relative mapping to rus

f

an automo rphism

f

an n-dimensional manifold with b

unda

ry

h

: (F, ∂F ) → (F, ∂F ) with h|∂F = id. is the closed (n + 1)-dimensional manifold

t

(h ) = T (h ) ∪∂F×S 1 ∂F × D 2 .

A

closed (n + 1)-dimensional manifold

M

has an

p

en b

k

decomp

sition

if

M

= t (h ) fo r some h .

Example

: fo r a b red knot k : Sn− 1 ⊂ Sn +1

Sn

+1 = t (h ), with h the mono dromy

f

the Seifert surface F n ⊂ Sn +1 , ∂F = k (Sn− 1 ).

Note

:

p

en b

k

= b re bundle

ver S

1 if

∂F

= ∅. 8

SLIDE 9 Brief histo ry

f
p

en b

ks
Alexander

(1923) : every 3-dimensional manifold has an

p

en b

k

decomp

sition.
Wink

elnk emp er (1972) : fo r n ≥ 7 a simply- connected n-dimensional manifold has an

p

en b

k

decomp

sition

if and

nly

if signature(M ) = 0 ∈ Z .

Quinn

(1979) : non-simply-connected

bstruction

theo ry in dimensions ≥ 5 to the existence and uniqueness

f
p

en b

k

decomp

sitions

: { asymmetric Witt

bstruction

in even dimensions, { no

bstruction

in

dd

dimensions. 9

SLIDE 10 Recognizing

p

en b

ks

homologically

M

= compact n-dimensional manifold.

A

= Z [π 1 (M )] , A[z, z− 1 ] = Z [π 1 (M × S 1 )] .

Theo

rem 3 If n ≥ 6 then M has an

p

en b

k

decomp

sition

if and

nly

if the − 1A[z, z− 1 ]-co ecient symmetric signature (= Quinn

p

en b

k
bstruction)

is

σ∗

(M ; − 1A[z, z− 1 ]) = 0 ∈ Ln (− 1A[z, z− 1 ]) .

A

manifold

f

dimension ≥ 6 has an

p

en b

k

decomp

sition

if and

nly

if it is π 1

b
rdant

to a b re bundle

ver S

1 . 10

SLIDE 11 Algeb raic K

theo

ry transversalit y

Co

dimension 1 geometric transversalit y : { every innite cyclic cover

f

a compact manifold has a compact fundamental domain.

Co

dimension 1 algeb raic transversalit y : { every nite f.g. free A[z, z− 1 ]-mo dule chain complex C has an algeb raic fundamental domain, with a chain equivalence

C

(f − zg : D [z, z− 1 ] → E [z, z− 1 ]) fo r A-mo dule chain maps f, g : D → E . (Higman (1940), W aldhausen (1972))

K

1 (A[z, z− 1 ]) = K 1 (A)⊕K (A)⊕ 2 Nil (A) . (Bass, Heller & Sw an (1965)) 11

SLIDE 12 Algeb raic P

inca

r

e

complexes

A

= ring with involution a → a .

An n-dimensional

algeb raic P

inca

r

e

complex

ver A

is an A-mo dule chain complex C with a symmetric chain equivalence

φ ≃ φ∗

: Cn−∗ = HomA (C, A)n−∗ → C inducing dualit y Hn−∗ (C ) ∼ = H∗ (C ).

Simila

rly fo r pairs, cob

rdism.
Ln

(A) = cob

rdism

group

f n-dimensional

algeb raic P

inca

r

e

complexes

ver A.
Simila

r description

f

W all surgery

bstruc-

tion groups Ln (A), using extra quadratic structure. Dierence in 2-p rima ry to rsion

nly

. 12

SLIDE 13 Symmetric signature

The

symmetric signature

f

an n-dimensional P

inca

r

e

space M is

σ∗

(M ) = (C (

M

), φ ) ∈ Ln (Z [π 1 (M )]) with

M

the universal cover

f M

and

φ

= [M ] ∩ − (Mishchenk

(1974)).
Homotop

y inva riant.

Bo

rdism inva riant

σ∗

: n (K ) → Ln (Z [π 1 (K )]) ; M → σ∗ (M ) .

Example

: fo r n = 4k

σ∗

(M ) = signature(M ) ∈ L 4k (Z ) = Z . 13

SLIDE 14 Algeb raic L

theo

ry transversalit y

The

asymmetric L

groups
f

a ring with involution A a re the cob

rdism

groups

f

pairs (C, λ ) with C an n-dimensional f.g. free A-mo dule chain complex and

λ

: Cn−∗ → C a chain equivalence.

The

asymmetric L

groups

a re fo r

dd n.
Theo

rem 4 The symmetric L

groups

Ln

(− 1A[z, z− 1 ])

f

F redholm lo calization (with z = z− 1 ) a re isomo rphic to the asymmetric L

groups
f A.

14

SLIDE 15 Manifold transversalit y

T

rue.

The

b

rdism

group n (X )

f

maps

Mn

= n-dimensional manifold → X is a generalized homology theo ry .

K

unneth fo rmula n (K × S 1 ) = n (K ) ⊕ n− 1 (K )

Co

dimension 1 transversalit y: every map

f

: Mn → K × S 1 homotopic to

ne

with

Nn−

1 = f− 1 (K × pt.) ⊂ Mn a co dimension 1 submanifold.

An

innite cyclic cover

f

a compact manifold has a compact manifold fundamental domain. 15

SLIDE 16 P

inca

r

e

space transversalit y

F

alse in general { same

bstructions

as fo r algeb raic P

inca

r

e

complex transversalit y .

A

nite n-dimensional P

inca

r

e

space P is a nite CW complex with

Hn−∗

(P ) ∼ = H∗ (P ) .

The

b

rdism

group h

n

(X )

f

maps P → X is not a generalized homology theo ry .

h

n

(K × S 1 ) = h

n

(K ) ⊕ p

n−

1 (K ) with p

∗

(K ) the b

rdism

group

f

nitely dominated P

inca

r

e

spaces with map to K .

An

innite cyclic cover

f

a nite P

inca

r

e

space has a nitely dominated fundamental domain. 16

SLIDE 17 References

Lo

w er K

and L
theo

ry LMS Lecture Notes 178, Camb ridge (1992)

Finite

domination and Novik

v

rings T

p
logy

34, 619{632 (1995)

The

b

rdism
f

automo rphisms

f

manifolds from the algeb raic L

theo

ry p

int
f

view Pro c. 1994 Bro wder Conference, Ann.

f

Maths. Studies 138, 314{327, Princeton (1996)

(with

B. Hughes) Ends

f

complexes T racts in Mathematics 123, Camb ridge (1996) 17