Morse homology

Morse homology

Morse homology

is a tool developed by Thom, Smale, and Milnor for

homology

theory.

Take

M

to be a smooth

compact manifold

. Throughout we assume that

f

is a suitable

Morse function

, that is, all

critical points

of

f

are

nondegenerate

. We must first make some definitions before defining the Morse homology.
Choose a Riemannian metric on

M

so that the notion of a

gradient

vector field

makes sense.
The map

ϕ

:

ℝ

×

M

→

M

such that

d

d

⁢

t

⁢

ϕ

⁢

(

t

,

x

)

=

-

∇

⁡

f

⁢

(

ϕ

⁢

(

t

,

x

)

)

,

with

ϕ

⁢

(

0

,

x

)

=

Id

, is called the

negative

gradient flow of

f

. Let

p

be a critical point of

f

, and define

W

p

s

:=

{

x

∈

M

|

lim

t

→

∞

⁡

ϕ

⁢

(

t

,

x

)

=

p

}

⁢

and

⁢

W

p

u

:=

{

x

∈

M

|

lim

t

→

-

∞

⁡

ϕ

⁢

(

t

,

x

)

=

p

}

to be the stable and unstable manifolds respectively.
Thom realized that one could decompose

M

into its unstable manifolds and arrive at something that is homologically equivalent to its

handle decomposition

, but this

decomposition

was not a

CW complex

, hence it was hard to say anything about the

homotopy type

of

M

. But Smale realized that if we impose more conditions on the metric itself, then we can make this into a CW complex.

The pair

(

f

,

g

)

, where

f

is a Morse function and

g

is the Riemannian metric, is called Morse-Smale pair, if for every pair

p

,

q

of critical points of

f

,

W

p

u

is

transverse

to

W

q

s

. This is known as the Morse-Smale condition.
This condition actually holds for a generic Riemannian metric on M. With this restriction, this makes Thom’s decomposition into a CW complex.

We can define a complex called the

Morse complex

as follows:

Let

Crit

k

⁡

(

f

)

be the set of critical points of

f

of index

k

. We define the chain group ,

C

k

⁢

(

f

)

to be the formal

linear combination

with

integer

coefficients of elements of

Crit

k

⁡

(

f

)

. We must also keep track of the signs of the

flow lines

. (However, it is true if you count mod 2, the Morse complex computes homology with coefficients in

Z

2

.)
To make this a

chain complex

we must define the differential map.
The map

δ

k

:

C

k

→

C

k

-

1

applied to a critical point

p

is a formal sum of critical points with

q

given by this number. It is possible to prove that

δ

2

=

0

, making this into a chain complex.

The homology of this complex is called the Morse homology. It can be shown to be

isomorphic

to the

singular homology

of

M

.

Note: There is another way of realizing the Morse homology using

Hodge theory

, an idea pioneered by Edward Witten. His idea is essentially to

conjugate

the

d

operator by

e

s

⁢

f

and it can be shown that this conjugation again leads to another

isomorphism

between the set of harmonic forms and the

De Rham cohomology

. This parameter

s

is like a curve of chain complexes and Witten claimed that if

s

is large enough, then we can obtain a space whose

dimension

is the number of critical points of a given index and the

boundary operator

induced on

d

is the number of critical paths between critical points, as before. Witten did not prove this idea rigorously, but it was done later by Helffer and Sjostrand.

Title

Morse homology

Canonical name

MorseHomology

Date of creation

2013-03-22 15:21:09

Last modified on

2013-03-22 15:21:09

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

17

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 58A05

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