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SLIDE 1

CIRCLE-VALUED MORSE THEORY AND NOVIKOV HOMOLOGY

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

T

raditional Mo rse theo ry deals with dierentiable real-valued functions

f

: M → R and

rdina

ry homology H∗ (M ).

Circle-valued

Mo rse theo ry deals with dierentiable circle-valued functions

f

: M → S 1 and Novik

v

homology HNov

∗

(M ). The circle-valued theo ry is new er and ha rder!

The

circle-valued theo ry has applications to the structure theo ry

f

non-simply-connected manifolds, dynamical systems, symplectic top

logy

, Flo er theo ry , Seib erg-Witten theo ry etc. 1

SLIDE 2 Novik

v
S.P

.Novik

v

(1938 {),

ne
f

the founding fathers

f

surgery theo ry .

Proved

the top

logical

inva riance

f

rational P

ntrjagin

classes fo r dierentiable manifolds (1965), fo r which he w as a w a rded the Fields Medal in 1970.

Last

pap er in surgery theo ry (1969) fo rmulated the Novik

v

conjecture.

Intro

duced circle-valued Mo rse theo ry in 1981, motivated b y physical p roblems in electro- magnetism and uid mechanics.

Autho

r

f

"T

p
logy"

(V

lume

12

f

Encyclop edia

f

Mathematical Sciences, Sp ringer, 1996) { the b est intro duction to high-dimensional manifold top

logy!

2

SLIDE 3 The p rogramme

The

geometrically dened Mo rse-Smale chain complex CMS (f )

f

a real-valued Mo rse function f : M → R is w ell-understo

d.

The geometrically dened Novik

v

chain complex CNov (f )

f

a circle-valued Mo rse function f : M → S 1 is not so w ell-understo

d.
Objective:

mak e the Novik

v

complex as w ell-understo

d

as the Mo rse-Smale complex! F eed algeb ra back into top

logy

.

The

strategy: lift f : M → S 1 to innite cyclic covers f : M → R and compa re

CNov

(f ) to CMS (fN ), with

fN

= f| : MN = f− 1 [0, 1] → [0, 1]

The

general theo ry w

rks

fo r a rbitra ry π 1 (M ). Will concentrate

n

the 'simply-connected' sp ecial case π 1 (M ) = Z , π 1 (M ) = {1}. 3

SLIDE 4 Real-valued Mo rse functions

A

critical p

int
f

a dierentiable function

f

: M → R is a zero p ∈ M

f ∇f

: τM → τR .

A

critical p

int p ∈ M

is nondegenerate if

f

(p + (x 1, x 2, . . . , xm )) = f (p ) −

i

j

=1 (xj ) 2 +

m

j

=i+1 (xj ) 2 nea r p with i the index

f f

. W rite Criti (f ) fo r the set

f

index i critical p

ints
f f

.

A

function f : M → R is Mo rse if every critical p

int

is nondegenerate. If M is compact and non-empt y then a Mo rse f : M →

R

has a nite numb er

ci

(f ) = |Criti (f )| ≥

f

critical p

ints

with index i. Note that

c

(f ) > 0, cm (f ) > (minimax p rinciple). 4

SLIDE 5 Where do real-valued Mo rse functions come from?

Nature

(= geometry)

Mo

rse functions f : M → R a re dense in the space

f

all dierentiable functions

n M

.

Mo

rse theo ry investigates the relationship b et w een the algeb raic top

logy
f M

and the Mo rse functions

n M

. T ypical p roblem: given M , what a re the minimum num- b er

f

critical p

ints
f

a Mo rse function

f

: M → R ? As usual, it is easier to nd answ er fo r dim (M ) ≥ 5. 5

SLIDE 6 Gradient

w
A

vecto r eld v : M → τM is gradient-lik e fo r a Mo rse function f : M → R if there exists a Riemannian metric ,

n M

with

v, w

= ∇f (w ) ∈ R (w ∈ τM ) .

A

do wnw a rd v

gradient
w

line γ : R → M satises

γ′

(t ) =

− v

(γ (t )) ∈ τM (γ (t )) (t ∈ R ) . A v

gradient
w

line sta rts at a critical p

int
f

index i lim

t→−∞

= p ∈ Criti (f ) and ends at a critical p

int
f

index i − 1 lim

t→∞

= q ∈ Criti− 1 (f ) . 6

SLIDE 7 Mo rse theo ry and surgery

A

critical value

f

Mo rse f : M → R is

f

(p ) ∈ R fo r critical p

int p ∈ M

. Can assume the critical values a re distinct, and that index (p ) ≤ index(p′ ) if f (p ) < f (p′ ).

W

rite Na = f− 1 (a ) ⊂ M fo r any regula r (= non-critical) value a ∈ R .

Theo

rem (Thom, 1949) (i) If f : M → [a, b ] has no critical values then (M ; Na, Nb ) ∼ = Na × ([0, 1]; {0}, {1}) . (ii) If f : M → [a, b ] has

nly
ne

critical value c ∈ [a, b ],

f

index i, then (M ; Na, Nb ) is the trace

f

surgery

n Si−

1×Dm−i ⊂ Na with

Nb

= (Na\Si− 1 × Dm−i ) ∪ Di × Sm−i− 1 ,

M

= Na × [0, 1] ∪ Di × Dm−i . 7

SLIDE 8

A

Mo rse function f : M → R determines a handleb

dy

decomp

sition
f M

M

=

m

i=0
ci

(f )

Di × Dm−i .

The Mo rse-Smale transversalit y condition

Theo

rem (Smale, 1962) F

r

every Mo rse

f

: M → R there is a class GT (f )

f

gradient- lik e vecto r elds v fo r f such that there is

nly

a nite numb er n(p, q )

f v
gradient
w

lines from p to q whenever index(q ) = index(p ) − 1 .

GT

(f ) is dense in the space

f

all gradient- lik e vecto r elds

n M

. 8

SLIDE 9 The Mo rse-Smale complex

The

Mo rse-Smale complex C = CMS (M, f, v ) fo r Mo rse f : M → R and v ∈ GT (f ) is a based f.g. free Z

mo

dule chain complex with Ci = Z [Criti (f )].

The

dierentials a re given b y the signed numb ers

f v
gradient
w

lines

d

: Ci → Ci− 1 ; p →

q∈Criti−1

(f )

n(p, q

)q .

The

Mo rse-Smale complex is the cellula r chain complex

f

the CW structure

n M

with

ne i-cell

fo r each critical p

int
f f
f

index i, CMS (M, f, v ) = C (M ), so

H∗

(CMS (M, f, v )) = H∗ (M ) .

Can

also dene CMS fo r Mo rse f : (M ; N, N′ ) → ([0, 1]; {0}, {1}), with CMS (M, f, v ) = C (M, N ). 9

SLIDE 10 The Mo rse inequalities

The

Betti numb ers

f

a nite CW complex

M

a re dened b y

bi

(M ) = dimZ (Hi (M )/Ti (M )) ,

qi

(M ) = minimum no. generato rs

f Ti

(M ) with

Ti

(M ) = {x ∈ Hi (M ) | nx = fo r some n = 0 ∈ Z} the to rsion subgroup

f Hi

(M ).

Theo

rem (Mo rse, 1927) The numb er ci (f )

f

index i critical p

ints
f

a Mo rse function

f

: M → R is b

unded

b elo w b y

ci

(f ) ≥ bi (M ) + qi (M ) + qi− 1 (M ) . Pro

f

A f.g. free Z

mo

dule chain complex

C

with H∗ (C ) = H∗ (M ) must have dimZ (Ci ) ≥ bi (M ) + qi (M ) + qi− 1 (M ) . In pa rticula r, this applies to C = CMS (M, f, v ). 10

SLIDE 11 The Mo rse inequalities a re sha rp fo r

π

1 (M ) = {1}

Theo

rem (Smale, 1962) An m

dimensional

manifold M with m ≥ 5 and π 1 (M ) = {1} admits a Mo rse function f : M → R with

ci

(f ) = bi (M ) + qi (M ) + qi− 1 (M ) .

Proved

b y handle cancellation.

The

situation is much mo re complicated fo r π 1 (M ) = {1}. Need algeb raic K

theo

ry

f

the Z [π 1 (M )]-mo dule version

f CMS

(M, f, v ) to give sha rp b

unds
n

minimum num- b er

f

critical p

ints
f

Mo rse f : M → R (Sha rk

).

11

SLIDE 12 Circle-valued Mo rse functions

A

critical p

int
f

a dierentiable function

f

: M → S 1 is zero p ∈ M

f ∇f

: τM → τS 1 .

A

critical p

int p ∈ M

is nondegenerate if

f

(p + (x 1, x 2, . . . , xm )) = f (p ) −

i

j

=1 (xj ) 2 +

m

j

=i+1 (xj ) 2 nea r p with i the index

f f

. A function f is Mo rse if every critical p

int

is nondegenerate.

If M

is compact and non-empt y then a Mo rse f : M → S 1 has a nite numb er

ci

(f ) ≥

f

critical p

ints

with index i.

Can

dene gradient-lik e v : M → τM , GT (f ) etc., as fo r the real-valued case. 12

SLIDE 13 Where do circle-valued Mo rse functions come from?

Nature,

cohomology , and knot theo ry .

Mo

rse functions f : M → S 1 a re dense in the space

f

all dierentiable functions

n

M

rep resenting xed c ∈ H 1 (M ) = [M, S 1 ].

T

ypical p roblem: given c ∈ H 1 (M ) what a re the minimum numb ers ci (f )

f

critical p

ints
f

a Mo rse function f : M → S 1 with

f∗

(1) = c ∈ H 1 (M )?

F
r m ≥

6 can apply the cancellation metho d

f

real-valued Mo rse theo ry , but the alge- b raic b

k-k

eeping is much ha rder.

Circle-valued

Mo rse theo ry extends to the Mo rse theo ry

f

closed 1-fo rms, rep resenting classes c ∈ H 1 (M ; R ). 13

SLIDE 14 Fib re bundles

ver S

1

The

mapping to rus

f

a map h : N → N is

T

(h ) = N × [0, 1]/{(x, 0) ∼ (h (x ), 1)} , with canonical p rojection

p

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

If h

: N → N is a dieomo rphism

f

a closed (m − 1)-dimensional manifold then T (h ) is a closed m

dimensional

manifold. The p rojection p : T (h ) → S 1 is a b re bundle , such that p− 1 (a ) ∼ = N fo r each a ∈ S 1 . The innite cyclic cover

f T

(h )

p∗R

= T (h ) = N × R is homotop y equivalent to N .

A

b re bundle f : M → S 1 is a Mo rse map with c∗ (f ) = 0. 14

SLIDE 15 Fib ering

bstruction

theo ry

If f

: M → S 1 is homotopic to b re bundle then M = f∗R is homotop y equivalent to a nite CW complex (= b re). Stallings (1962): pa rtial converse fo r 3-manifolds M .

Bro

wder-Levine (1965) : fo r m ≥ 6 a function f : Mm → S 1 with f∗ : π 1 (M ) ∼ = Z is homotopic to b re bundle if and

nly

if

M

= f∗R is homotop y equivalent to a nite

CW

complex.

F

a rrell (1967) and Sieb enmann (1970) : fo r m ≥ 6 a function f : M → S 1 is homotopic to the p rojection

f

a b re bundle if and

nly

if M nitely dominated and a Whitehead group

bstruction

(M ) ∈

Wh

(π 1 (M )) is (M ) = 0.

Proved

b y handle cancellation and exchanges. 15

SLIDE 16 The Novik

v

ring

The

ring Z [[z ]] consists

f

the p

w

er series

p

(z ) =

∞

j

=0

njzj

(nj ∈ Z ) . Note that p (z ) ∈ Z [[z ]] is a unit if and

nly

if p (0) = n 0 ∈ Z is a unit (= ±1). Example: 1 − z .

The

Novik

v

ring

Z

((z )) = Z [[z ]][z− 1 ] consists

f

the p

w

er series

∞

j

=−∞

njzj

with co ecients nj ∈ Z such that fo r some k ∈ Z

nj

= fo r j < k . 16

SLIDE 17 The real-valued lift

f

a circle-valued Mo rse function

Given

Mo rse f : M → S 1 , v ∈ GT (f ) lift to Mo rse f : M → R , v ∈ GT (f ). Lift each

p ∈

Criti (f ) to p ∈ Criti (f ).

Cho
se

the generating covering translation

z

: M → M to b e the

ne

pa rallel to v :

M → τM

, dz, v > 0. In the universal example

z

: S 1 = R → R ; t → t − 1 .

F
r

any p ∈ Criti (f ), q ∈ Criti− 1 (f ) let

k

= [f (p ) − f (q )] ∈ Z . The signed numb ers nj = n(p, zjq ) ∈ Z

f

v

gradient
w

lines a re such that

nj

= fo r j < k . 17

SLIDE 18 The Novik

v

complex

The

Novik

v

complex C = CNov (M, f, v ) fo r Mo rse f : M → S 1 and v ∈ GT (f ) is dened geometrically to b e the based f.g. free Z ((z ))-mo dule chain complex with

Ci

= Z ((z ))[Criti (f )] .

The

dierentials a re given b y the signed numb ers

f v
gradient
w

lines

d

: Ci → Ci− 1 ; p →

q∈Criti−1

(f )

n(p, zjq

)zjq .

Example CNov

(M, f, v ) = fo r b re bundle.

Exercise

W

rk
ut CNov

(S 1, f, v ) fo r

f

: S 1 → S 1 ; [t ] → [4t−9t 2 +6t 3 ] (0 ≤ t ≤ 1) . 18

SLIDE 19 Novik

v

homology

The

Novik

v

homology

f

a nite CW complex M with a map f : M → S 1 is dened b y

HNov

∗

(M, f ) = H∗ (Z ((z )) ⊗Z[z,z−1 ] C (M )) with M = f∗R . The Novik

v

homology dep ends

nly
n

the cohomology class

c

= f∗ (1) ∈ [M, S 1 ] = H 1 (M ) .

Theo

rem F

r

any map f : M → S 1

n

a

nite CW

complex M the Novik

v

homology is HNov

∗

(M, f ) = if (and fo r π 1 (M ) = {1}

nly

if ) M is homotop y equivalent to a

nite CW

complex.

Example

If f : T (2 : S 1 → S 1 ) → S 1 is the canonical p rojection then

HNov

∗

(T (2), f ) = Z ((z ))/(2−z ) =

Q

2 = 0 . 19

SLIDE 20 The Novik

v

complex has Novik

v

homology

Theo

rem (Novik

v,

1982) The Novik

v

complex CNov (M, f, v )

f

a Mo rse function f :

M → S

1 is chain equivalent to Z ((z ))⊗Z[z,z−1 ]

C

(M ), so that

H∗

(CNov (M, f, v )) ∼ = HNov

∗

(M, f ) .

The

Novik

v

complex is directly constructed from f : M → S 1 .

The

Novik

v

homology uses the structure

f M

as a CW complex, which in general will have many mo re cells than there a re critical p

ints

in f . 20

SLIDE 21 The Mo rse-Novik

v

inequalities

The

Novik

v

numb ers

f

a nite CW complex M with f ∈ H 1 (M ) a re dened b y

bNov

i

(M, f ) = dimZ((z )) (HNov

i

(M, f )/T Nov

i

(M, f )) ,

qNov

i

(M, f ) = min. no.

f

generato rs

f T Nov

i

(M, f ) with

T Nov

i

(M, f ) = {x ∈ HNov

i

(M, f ) |

nx

= fo r some n = 0 ∈ Z ((z ))} the to rsion Z ((z ))-submo dule

f HNov

i

(M, f ).

Theo

rem (Novik

v,

1982) The numb er ci (f )

f

index i critical p

ints
f

a Mo rse function f : M → S 1 is b

unded

b elo w b y

ci

(f ) ≥ bNov

i

(M, f )+qNov

i

(M, f )+qNov

i−

1 (M, f ) . Pro

f

Since Z ((z )) is a p rincipal ideal domain, a f.g. free Z ((z ))-mo dule chain complex C with H∗ (C ) = HNov

∗

(M, f ) must have dimZ((z )) (Ci ) ≥ bi (M, f )+qi (M, f )+qi− 1 (M, f ) . 21

SLIDE 22 The Mo rse-Novik

v

inequalities a re sha rp fo r π 1 (M ) = Z

Theo

rem (F a rb er, 1985) An m

dimensional

manifold M with m ≥ 6 and π 1 (M ) = Z admits a Mo rse function f : M → S 1 with

ci

(f ) = bNov

i

(M, f )+qNov

i

(M, f )+qNov

i−

1 (M, f ) .

Proved

b y handle cancellation and handle exchanges.

The

situation is much mo re complicated fo r π 1 (M ) = Z . Need algeb raic K

theo

ry

f

the

Z

[π 1 (M )]-mo dule version

f CNov

(M, f, v ) to give sha rp b

unds
n

minimum numb er

f

critical p

ints
f

Mo rse f : M → S 1 , with

Z

[π 1 (M )] the Novik

v

completion

f

Z

[π 1 (M )] (P ajitnov). 22

SLIDE 23 Geometric fundamental domains

Given

Mo rse f : M → S 1 and regula r value

a ∈ S

1 lift to a ∈ R . Cut M along f− 1 (a ) =

N ⊂ M

to get fundamental domain (MN ; N, z− 1N ) = f− 1 ([a, a +1]; {a}, {a +1}) fo r the innite cyclic cover

M

= f∗R =

∞

j

=−∞

zjMN .

The

restriction

fN

= f| : (MN ; N, z− 1N ) → ([a, a +1]; {a}, {a +1}) is a real-valued Mo rse function with the same numb ers

f

critical p

ints

as f

ci

(fN ) = ci (f ) .

The

Mo rse theo ry

f

circle-valued f is the Mo rse theo ry

f

real-valued fN fo r all p

ssible

choices

f N

. 23

SLIDE 24 Handle exchanges

Supp
se

given a map f : M → S 1

n

an m

dimensional

manifold M and a fundamental domain (MN ; N, z− 1N ) fo r M = f∗R , with

N

= f− 1 (a ) fo r a regula r value a ∈ S 1 .

A

handle exchange uses an emb edding (Di × Dm−i, Si− 1 × Dm−i ) ⊂ (MN\z− 1N, N ) to

btain

another fundamental domain (MN′ ; N′, z− 1N′ ) fo r M b y

N′

= (N\Si− 1 × Dm−i ) ∪ Di × Sm−i− 1 ,

MN′

= (MN\Di × Dm−i ) ∪ z− 1 (Di × Dm−i ) . Any t w

fundamental

domains fo r M a re related b y a sequence

f

handle exchanges. 24

SLIDE 25 Handle cancellation

Given f

: M → S 1 and a choice

f

fundamental domain (MN ; N, z− 1N ) can try to cancel as many handle pairs in fN : MN →

R

as p

ssible.

Handle cancellations co r- resp

nd

to homotopies f ≃ f′ to another Mo rse function f′ : M → S 1 with few er critical p

ints,

k eeping N = f− 1 (a ) ⊂ M xed.

In
rder

to decide if there exists a homotop y f ≃ f′ to a Mo rse f′ with few er critical p

ints

need to have algeb raic description

f

all p

ssible

choices

f N

.

The

algeb raic theo ry

f

surgery has a de- pa rtment dealing with the algeb raic theo ry

f

handle exchanges. 25

SLIDE 26 The algeb raic construction

f

the Novik

v

complex (I)

The

Novik

v

complex can b e constructed algeb raically from the Mo rse-Smale complex

f

a fundamental domain.

Given

Mo rse f : M → S 1 , v ∈ GT (f ), a regula r value a ∈ S 1 , let N = f− 1 (a ) ⊂ M . Let (MN ; N, z− 1N ) b e the co rresp

nding

fundamental domain fo r M = f∗R with Mo rse

fN

= f| : MN → R , vN = v| ∈ GT (fN ).

The

handleb

dy

structure

MN

= N × [0, 1] ∪

m

i=0
ci

(f )

Di × Dm−i

gives (MN, N ) the structure

f

a relative

CW

pair with ci (f ) i-cells. 26

SLIDE 27 The algeb raic construction

f

the Novik

v

complex (I I)

Given CW

structure

n N

with ci (N ) i-cells

btain CW

structures

n MN

with

ci

(MN ) = ci (N ) + ci (f ) i-cells and a CW structure

n M

with

ci

(M ) = ci (N ) + ci− 1 (N ) + ci (f ) i-cells.

Let g

: C (N ) → C (MN ) b e the inclusion

f

chain complexes induced b y N ⊂ MN which is the inclusion

f

a sub complex. Let

h

: C (z− 1N ) → C (MN ) b e the chain map induced b y the inclusion z− 1N ⊂ MN which is not the inclusion

f

a sub complex.

The

cellula r chain complex

f M

is the algeb raic mapping cone C (M ) = C (φ )

f

the

Z

[z, z− 1 ]-mo dule chain map

φ

= g−zh : C (N )[z, z− 1 ] → C (MN )[z, z− 1 ] 27

SLIDE 28 The algeb raic construction

f

the Novik

v

complex (I I I)

The Z

[z, z− 1 ]-mo dule chain map φ induces a Z ((z ))-mo dule chain map

φ

= g − zh : C (N )((z )) → C (MN )((z )) which is a split injection in each degree, with contractible k ernel (= algeb raic mo del fo r closed v

gradient
w

lines in M ).

Theo

rem The Novik

v

complex

f

a Mo rse

f

: M → S 1 fo r app rop riate v ∈ GT (f ) is

CNov

(M, f, v ) = cok er (

φ ) .

The p rojection

C

M

; Z ((z ))

= C

(

φ

)

→ CNov

(M, f, v ) = cok er (

φ )

is a chain equivalence: the v

gradient
w

lines in M a re pieced together from the w a y they cross zjMN ⊂ M (j ∈ Z ). 28