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Floer Homology

P B Kronheimer, Harvard University,

Cambridge, MA, USA

Introduction

Morse theory allows one to reconstruct the homology

of a compact manifold B from data obtained from the

gradient flow of a function f : B !R, the Morse

function. The term ‘‘Floer homology’’ is used to

describe homology groups that arise from carrying

out the same construction, but in a setting where the

space B is replaced by an infinite-dimensional mani-

fold (a space of maps, or a space of configurations for a

gauge theory), and where the gradient trajectories of

the Morse function correspond to solutions of an

elliptic differential equation. There are two important

types of such homology theories that have been

extensively developed, and the study of both was

initiated in the 1980s by Andreas Floer. In the first

type, the elliptic equation that arises is a Cauchy–

Riemann equation, whose solutions are pseudoholo-

morphic maps from a two-dimensional domain into a

symplectic manifold. In the second type, the elliptic

equation is an equation of gauge theory on a

4-manifold: either the anti-self-dual Yang–Mills

equations or the Seiberg–Witten equations. Important

antecedents of Floer’s work included work of Conley,

Zehnder, and others on the symplectic fixed-point

problem, and Witten’s ideas about Morse theory.

This article describes the background material

from Morse theory before discussing Floer homol-

ogy of Cauchy–Riemann type and its application to

the Arnol’d conjecture in symplectic topolog y. Floer

homology in the context of four-dimensional gauge

theories is discussed more briefly.

Morse Theory

Let B be a smooth, compact manifold and f : B !R

a smooth function. A critical point p of f is said to

be nondegenerate if the Hessian of f is a nonsingular

operator on T

B. The function f is a Morse function

if all its critical points are nondegenerate. In the

presence of a Riemannian metric g on B, the

derivative df becomes a vecto r field, the gradient

rf , and we can consider the down ward gradient-

flow equation for a path x(s)inB:

¼rf ðxÞ

If p and q are nondegenerate critical points, let us

write M(p, q) for the space of solutions x(s)

satisfying

lim

s!1

xðsÞ¼p

lim

s!þ1

xðsÞ¼q

To understand the structure of M(p, q), consider the

linearization of the gradient-flow equation at a

solution x 2 M(p, q). This is a linear equation for a

vector field X along the path x in B, and takes the

form

@=@s

X ¼ rrf ðXÞ½1

where rrf is the covariant deriva tive of the

gradient rf , an operator on tangent vectors. Let 

be the dimension of the space of solutions X to this

linear equation, with the boundary conditions

lim

s !1

X(s) = 0, and let 

be the dimension of

the space of solutions to the adjoint equation

@=@s

X ¼ þrrf ðXÞ

We say that the trajectory x is ‘‘regular’’ if 

= 0. In

this case, the trajectory space M(p, q) has the

structure of smooth manifold near x: its dimension

is 

and its tangent space is the space of solutions X

to [1]. The gradient flow is said to be Morse–Smale

if all trajectories between critical points are regular.

If f is any Morse function, one can always choose

the metric g so that the corresponding flow is

Morse–Smale. (It is also the case that one can leave

g fixed and perturb f to achieve the same effect.)

356 Floer Homology

In the Morse–Smale case, each M(p, q)isa

smooth manifold. The dimension of M(p, q) in the

neighborhood of a trajectory x depends only on

p and q, not otherwise on x. Indeed, even without

the regularity condition, the index of eqn [1],

namely the difference 

 

, is given by



 

¼ indexðpÞindexðqÞ

where index(p) denotes the number of negative

eigenvalues (counting multiplicity) of the Hessian

at p. In the Morse–Smale case therefore, the

dimension of M(p, q) is given by index(p) 

index(q). If x(s) is a solution of the gradient-flow

equation, then so is the reparametrized trajectory

x(s þ c); and this is different from x(s) as long as

p 6¼ q. Let us denote by



M(p, q) the quotient of

M(p, q) by the action of R given by these reparame-

trizations. We have

dim



Mðp; qÞ¼indexðpÞindexðqÞ1 ðp 6¼ qÞ

as long as the trajectory space is nonempty.

Let F

denote the field with two elements. The

Morse complex of a Morse–Smale gradient flow,

with coefficients in F

, is defined as follows. For

each i, let C

(f ) be the finite-dimensional vector

space over F

having a basis

; ...; e

indexed by the critical points p

, ..., p

with index i.

For each pair of critical points p and q with indices

i and i  1 respectively, let 

2 F

denote the

number of points in the zero-dimensional manifold



M(p, q), counted mod 2:



¼ #



Mðp; qÞðmod 2Þ

The Morse–Smale condition ensures that the zero-

dimensional space



M(p, q) is finite, so this defi nition

is satisfactory. Define a differe ntial

 : C

ðf Þ!C

i1

ðf Þ

ðe

Þ¼

indexðqÞ¼i1



The first important fact is that  really is a

differential: as long as the flow is Morse–Smale,

we have

the composite    : C

ðf Þ!C

i2

ðf Þ is zero ½2

We can therefore construct the homology of the

complex (C



(f ), ). This is the Morse homology:

ðf Þ¼

ker  : C

ðf Þ!C

i1

ðf ÞðÞ

im  : C

iþ1

ðf Þ!C

ðf ÞðÞ

½3

The proof of [2] is as follows. Suppose that p has

index i and r is a critical point with index i  2, and

consider



M(p, r ), which has dimension 1. The key

step is to understand that



M(p, r) is noncompact,

and that its ends correspond to ‘‘broken trajec-

tories’’: pairs (x

, x

) (modulo reparametrization),

where x

is a gradient traject ory from p to some q of

index i  1, and x

is a trajectory from q to r. The

number of ends is thus



. Since the number

of ends of a 1-manifold is even, this sum is zero in

. This sum is also the matrix entry of    from e

to e

;so   = 0.

The main result about Morse homology in finite

dimensions is the following:

Theorem 1 The Morse homology H

(f ) is iso-

morphic to the ordinary homology of the compact

manifold B with coeffic ients F

: the group H

(B; F

This result can be proved by first showing that

(f ) depends only on B, not on the choice of f or

the metric. (This step can be accomplished by

examining a nonau tonomous flow of the form

dx=ds = rf (s, x).) Then one can examine the

Morse complex in the case of a self-indexing

Morse function (where the value of f at the critical

points is a monotone-increasing function of their

index). In the self-indexing case, the unstable

manifolds of the critical points give rise to a cell

decomposition of the manifold B, and the Morse

complex is easily identified with the cellular chain

complex for this cell decomposition.

The sum of the dimensio ns of the Morse

homology groups cannot be larger than the sum of

the dimensions of the chain groups C

(f ), which is

the total number of critical points. The above

theorem therefore implies the following basic ver-

sion of the ‘‘Morse inequalities’’:

Corollary 2 The number of critical points of a

Morse function f : B !R cannot be less than

dim H

(B; F

The Morse complex can be refined in various

ways. For example, one can use integer coefficients

in place of coefficients F

by taking account of

orientations of the spaces of trajectories. One can

also introduce Morse theory with coefficients in a

local system, and in both these cases a version of the

above theorem continues to hold. One can also

study the Morse complex of a multivalued Morse

function: that is, one can start with closed 1-form 

on B, with nontrivial periods, and study the flow

generated by the corresponding vector field g

1

.

Such a theory was developed by Novikov.

The Morse complex can be generalized in a

different direction, replacing f by a functional

Floer Homology 357

related to a geometric problem. The canonical

example of this (and one of the very few cases in

which the theory works as in the finite-dimensional

case) is the case that B = LW is the space of loops

u : S

!W in a Riemannian manifold W and f is the

‘‘energy function,’’ f

(u) =

(du=dt)

dt . If the

Morse–Smale condition holds, then the Morse

homology H

) computes the homology of LW,

as expected. Critical points of f

are geodesics, and

the relationship between geodesics and the topology

of LW , for which Corollary 2 provides a prototype,

is an idea with many applications.

For the energy functional, the downward gradient-

flow equation is a parabolic equation (the ordinary

heat equation if the target space is Euclidean), and

a solution to the flow exists for each choice of

initial condition. Floer homology can be loosely

characterized as the Morse theory of certain

variational problems for which the gradient-flow

equation is not parabolic, but elliptic of first order:

the important models ar e the Cauchy–Riemann

equation in dimension 2, the anti-self-dual Yang–

Mills equations in dimension 4, or the closely

related Seiberg–Witten equations. For an elliptic

equation, one does not expect to solve the Cauchy

problem with arbitrary initial c ondition; so with

Floer homology, one is studying a functional for

which the gradient flow is not everywhere defined.

However, to define the Morse complex, the import-

ant thing is only that we have a good understanding

of the trajectory spaces M(p, q), w hich will now be

solution spaces for an elliptic problem of geometric

origin. The proof of Theorem 1 depends very much

on the fact that the flow is everywhere defined: this

theorem will therefore f ail for the Morse complexes

arising i n Floer theory, and one must look else-

where for a m eans to compute the Morse homology

groups.

Before discussing Floer homology in more specific

terms, we shall describe the problem in symplectic

geometry that motivated its development.

The Arnol’d Conjecture

A symplectic mani fold of dimension 2n is a smooth

manifold W equipped with a 2-form ! which is

closed and nondegenerate. On a symplectic mani-

fold, one can associate to each smooth function

H : W !R a vector field X

on W: the vector field

is characterized by the property that

!ðX

; VÞ¼dHðVÞ

for all vector fields V. In this situation, one refers to

H as the Hamiltonian and X

as the corresponding

Hamiltonian vector field. If W is compact, or if X

is otherwise complete, then this vector field gener-

ates a flow 

: W !W(t 2 R). We also wish to

consider the case that H is time dependent: we

suppose that H

: W !R is a Hamiltonian which

varies smoothly with t 2 R and is periodic, in that

tþ1

= H

. In this case, there is a time-dependent

Hamiltonian vector field X

, and we can con sider

the flow 

that it generates: so for x 2 W, the path



(x) will be the solution to



ðxÞ¼X

ðxÞ½4

with initial condition 

(x) = x. The Arnol’d con-

jecture, in one formulation, concerns the 1-periodic

solutions to this equation, or equivalently the fixed

points of 

: W !W. A fixed point x with 

(x) = x

is called nondegenerate if d

: T

X !T

X does not

have 1 as an eigenvalue. With this understood, one

version of the conjecture states:

Conjecture 3 Suppose W is compact and let H

any 1-periodic, time-dependent Hamiltoni an. If the

fixed points of 

are all nondegenerate, then the

number of fixed points is not less than the sum of

the Betti numbers of the manifold W.

There is another, more general version of this

conjecture. Let L  W be a closed Lagrangian

submanifold: that is, an n-dimensional submanifold

such that the restriction of ! to L as a 2-form is

identically zero. Le t L

 W be another Lagrangian,

obtained from L by a Hamiltonian isotopy: that is,

is 

(L), for some flow 

generated by a time-

dependent Hamiltonian H

as above.

Question 4 If L and L

intersect transversely, is it

always true that the number of intersection points of

L and L

is at least the sum of the Betti numbers of

the manifold L:

#ðL \ L

Þ

rankH

ðLÞ?

This is phrased as a question rather than a

conjecture, because the answer is certainly ‘‘no’’ in

some cases. For example, L mi ght be a circle

contained in a small disk in a symplectic 2-manifold,

in which case there is no reason why 

should not

move the disk to be completely disjoint from itsel f.

Nevertheless, with extra hypotheses, it is known

that the answer is often ‘‘yes.’’

We can exhibit Conjecture 3 as a special case of

Question 4, as follows. Given a symplectic manifold

(V, !), we can form the product W = VV, with the

symplectic form !

= p



! þ p



!, where the p

are

the two projections. The result of this definition is

358 Floer Homology

that the diagonal in V  V is a Lagrangian

submanifold,

L  W ¼ V  V

for this symplectic form. Let H

be a time- dependent

Hamiltonian on V, and let 

: V !V be the flow.

Then H

 p

is a time-dependent Hamiltonian

generating a flow on W. For the flow on W, the

image L

of the diagonal L  W at time 1 is the

graph of 

: V !V. Thus, (L \ L

) can be identified

with the set of fixed points of 

in V,andan

affirmative answer to Question 4 for L  W implies

Conjecture 3 for V.

Conjecture 3 and Question 4 can both be

extended to the case of isolated degenerate fixed

points of 

for Conjecture 3, or to the case of

isolated, nontransverse intersections for Question 4.

For example, one can ask whether, in the non-

transverse case, the sum of the intersection multi-

plicities can ever be less than the sum of the Betti

numbers.

Morse Theory and the Arnol’d Conjecture

The Arnol’d conjecture, and the related Question 4,

can both be studi ed by reformulating them as

questions about the number of critical points of a

carefully chosen functional.

We begin with the situation addressed by Con-

jecture 3. For simplicity, we suppose that 

(W)is

zero. Let B be the space of smooth, null-homotopic

loops in W:

B¼fu : S

!Wju is smooth and null homotop ic g

This is a smooth, infinite-dimensional manifold.

There is a natural functional f

: B!R, the sym-

plectic action, defined as

ðuÞ¼



ð!Þ

where v : D

!W is any extension of the map

u : S

!W. The extension v exists because u is null

homotopic, and the value of f

is independent of the

choice of v because 

(W) = 0. This functional can be

modified in the presence of a periodic Hamiltonian.

Introduce a coordinate t on S

with period 1, and so

regard u as a periodic function of t. Write the

Hamiltonian as H

as before, and define

f ðuÞ¼f

ðuÞþ

ðuðtÞÞdt

To compute the first variation of f, consider a one-

parameter family of loops u

(t) = u(s, t) parametrized

by s 2 R. We compute

f ðu

Þ¼

;



dt þ



;

 X

ðuÞ



using the relationship between dH

and X

. Thus, a

loop u 2Bis a critical point of f : B!R if and only

if it is a solution of the equation

¼ X

ðuðtÞÞ ½5

This means that there is a one-to-one correspon-

dence between these critical points and certain

1-periodic solutions of eqn [4]: these in turn

correspond to fixed points p of 

with the

additional property that the path 

(p) from p to

p is null homotopic.

To consider the formal gradient flow of the

functional f, on must introduce a metric on B.A

Riemannian metric g on the symplectic manifold

(W, !) is compatible with ! if there is an almost-

complex structure J : TW !TW such that

!(X, Y) = g(JX, Y) for all tangent vectors X and Y

at any point of W. Let g

be a 1-perio dic family of

compatible Riemannian met rics on W. Using these,

on can define an inner product on the tangent

bundle of B by the formula

hU; Vi¼

ðUðtÞ; VðtÞÞdt

in which U and V are tangent vectors a t u 2B,

regarded as vector fields along the loop u in W.We

can rewrite the above formula for the variation of

f in terms of this inner product:

; J

 X

ðuÞ



where J

is the almost-complex structure corre-

sponding to g

. Formally then, a one-parameter

family of loops u(s, t) is a solution of the downward

gradient-flow equations for the functional f with

respect to this metric, if u satisfies the differential

equation

þ J

 X

ðuÞ



¼ 0 ½6

In the absence of the term X

, and with W replaced

by C

with the standard J, this equation becomes the

Cauchy–Riemann equation du=d



z = 0, for a function

u of the complex variable z = s þ it, periodic in t.

Floer Homology 359

Let us now suppose we are in the situation of

Conjecture 3,soW is closed, and the fixed points of



are nondegenerate. As we have seen, each fixed

point p of 

corresponds to a 1-periodic solution u

of eqn [5] , a critical point of f. For each pair of fixed

points p and q, introduce M(p, q) as the space of

solutions of the formal gradient-flow equations of f,

running from p to q: that is, M(p, q) is the space of

maps u : R  S

!W satisfying eqn [6], with

lim

s!1

uðs; tÞ¼u

ðtÞ

lim

s!þ1

uðs; tÞ¼u

ðtÞ

With these definitions in place, one can follow the

same sequence of steps that we outlined previously

in the context of finite-dimensional Morse theory, to

construct the Morse complex. First, if u belongs to

M(p, q), we can consider the linearization at u of

eqns [6], to obtain the counterpart of eqn [1]. These

are linear equations for a vector field U(s, t) along u

in W, and take the form

@=@s

U þ J

@=@t

U þ hðUÞ¼0 ½7

where h is a linear operator of order zero. Let 

denote the dimension of the space solutions U which

decay at s = 1, and let 

denote the dimension of

the space of solutions of the formal adjoint

equation. Elliptic theory for the Cauchy–Riemann

equation, and the nondegeneracy condition for u

and u

, mean that the operator that appears on the

left-hand side of the equation is Fredholm: so both



and 

are finite, and the index 

 

deformation invariant. This index depends only on

p and q: we give it a name,



 

¼ indexðp; qÞ

As before, u is said to be regular if 

is zero. For

suitable choice of the almost-complex structures J

(or equivalently the metrics g

), the Morse–Smale

condition will hold: that is, the trajectories in all

spaces M(p, q) are regular. In this case, each M(p, q)

is a smooth manifold and has dimension index(p,q)

if it is nonempty.

The ‘‘relative index’’ index(p, q) plays the role of

the difference of the Morse indices in the finite-

dimensional case. It can be defined whether or not

M(p, q) is empty by considering an equation such as

[7] along an arbit rary path u(s, t). In general, there is

no natural way to define the ‘‘index’’ of p:ifwe

wish, we can select one fixed point p

and declare it

to have index zero; we can then define index(p)as

index(p, p

). Alternatively, we can regard the critical

points as indexed by an affine copy of Z (without a

preferred zero).

Imitating the construction of the Morse complex,

we define a vector space CF



over F

as having a

basis consisting of elements e

indexed by the fixed

points p. We then define  :CF



!CF



e

indexðp;qÞ¼1



where 

is defined by counting points in



M(p, q)as

before. The vector space CF



is Z-graded if we make

a choice of critical point p

to have index zero;

otherwise, CF



has an ‘‘affine’’ Z-gradi ng. The map

 maps CF

into CF

i1

To show that  is well defined, and to show that

   = 0, one must show that the zero-dimensional

spaces



M(p, q) are compact, and that the ends of

the one-dimensional spaces



M(p, r ) correspond

bijectively to broken trajectories, as in the finite-

dimensional case. Both of these desired properties

hold, under the Morse–Smale conditions; but this is

a very special feature of the specific problem.

Without the hypothesis that 

(W) is zero, addi-

tional noncompactness can arise from the following

‘‘bubbling’’ phenomenon. There could be a

sequence of solutions u

2 M(p, q)toeqns [6], and

a point (s

, t

)inR  S

, such that for suitable

constants 

converging to zero, the rescaled

solutions

ð; Þ¼u

ðs

þ 

; t

þ 

Þ

converge on compact subsets of the plane R

to a

nonconstant pseudoholomorphic map

u : CP

!W,

or more precisely a solution of the equation

@

þ J

@



¼ 0

(In the original coordinates, the derivatives of the u

would grow like 1=

near (s

, t

).) A pseudoholo-

morphic sphere always has nontrivial homology

class (and therefore nontrivial homotopy class); so

this sort of noncompactness does not occur when



(W) = 0.

Granted the compactness results, the proof that

   = 0 runs as before, and we can construct a Floer

homology group,



¼ ker ðÞ=imðÞ

Unlike the Morse homology of the energy func-

tional, the Floer homology does not yield the

ordinary homology of B. To compute it, one first

shows that it depends only on the symplectic

manifold (W, !), not on the choice of Hamiltonian

or metrics g

: this step is similar to the proof that

the finite-dimensional Morse homology H



(f ) does

not depend on the Morse function. Once one has

360 Floer Homology

established this independence, HF



can be computed

by examining a special case. Floer did this by taking

the Hamiltonian to be independent of t and equal to

a small negative multiple h of a fixed Morse

function h : W !R on the symplectic manifold. If

the multiple  2 R is small enough, the only fixed

points of 

are the stationary points of the flow,

and these are exactly the critical points of h.

Furthermore the only index-1 solutions of eqn [6]

for small  are the solutions u(s, t) with no t

dependence; and these are the solutions of

du=ds = rh, the downward gradient flow of h,

scaled by . In this case therefore, the Floer complex



is precisely the Morse complex C



(h) of the

Morse function h, and Theorem 1 yields:

Theorem 5 For a periodic, time-dependent Hamil-

tonian H

on a closed symplectic manifold (W, !)

with 

(W) = 0, the Floer homology HF



is iso-

morphic to the ordinary homology of W with F

coefficients, H



(W; F

Because the generators of CF



correspond to fixed

points p of 

such that the path 

(p) is null

homotopic, the number of these fixed points is not

less than the dimension of HF



, and therefore not

less than

dim H

(W; F

) because of the above

result. The sum of the mod 2 Betti number s is at

least as large as the sum of the ordinary Betti

numbers (the dimensions of the rational homol ogy

groups); so one deduces, following Floer,

Corollary 6 The Arnol’d conjecture (Conjecture 3)

holds for symplectic manifolds (W, !) satisfying the

additional condition 

(W) = 0.

Orientations can be introduced rather as in the

case of finite-dimensional Morse theory, allo wing

one to define Floer groups with arbitrary

coefficients.

The Arnol’d conjecture is now known to hold in

complete generality, without the hypothesis on 

The proof has been achieved by successive exten-

sions of the Floer homology technique. When 

(W)

is nonzero, the space B is not simply connected. The

first complication that arises is that the symplectic

action functional f

, and therefore f also, is multi-

valued. This is not an obstacle initially, because rf

is still well defined, and the spaces M(p, q)of

gradient trajectories can still be assumed to satisfy

the Morse–Smale condition: this is the type of

Morse theory considered by Novikov, as mentioned

above. Because 

(B) is nontrivial, M(p, q) is a union

of parts M

(p, q), one for each homotopy class of

paths from p to q. For each homotopy class z,we

have the index index

(p, q), which is the dimension

of M

(p, q).

The spaces M

(p, q) may now have additional

noncompactness, due to the presence of pseudo-

holomorphic spheres

u : CP

!W. The simplest

manifestation is when a sequence u

in M

(p, q)

‘‘bubbles off’’ a single such sphere at a point (s

, t

and converges elsewhere to a smooth trajectory u

(p, q), belonging to a different homotopy class.

Let  be the homology class of the sphere

u. Because

the sphere has positive area, the pairing of  with

the de Rham class [!] is positive: h[!], i> 0. The

indices are related by

index

ðp; qÞ¼index

ðp; qÞ2hc

ðWÞ;i

where c

(W) 2 H

(W; Z) is the first Chern class of a

compatible almost-complex structure. The symplec-

tic manifold is said to be ‘‘monotone’’ if, in real

cohomology, c

(W) is a positive multiple of [!]. In

the monotone case, we always have index

(p, q) <

index

(p, q), and no bubbling off can occur for

trajectory spaces M

(p, q) of index 2 or less: the

above formula either makes M

(p, q) a space

of negative dimension (in which case it is empty)

or a zero-dimensional space (in which case one

has to exploit an additional transversality argum ent,

to show that the holomorphic spheres belonging

to classes  with hc

(W), i= 1 cannot intersect one

of the loops u

in W). Since the construction of



involves only the trajectories of indices 1 and 2,

the construction goes through with minor changes.

Because index

(p, q) depends on the path z,

the group HF



will no longer be Z-graded: the

grading is defined only modulo 2d, where d is the

smallest nonzero value of hc

(W), i for spherical

classes .

In the case that W is not monotone, additional

techniques are needed to deal with the essential

noncompactness of the trajectory spaces. These

techniques involve (amongst other things) multi-

valued perturbations on orbifolds – a strategy that

requires the use of rational coefficients in order to

perform the necessary averaging. For this reason, in

the monotone case, the Arnol’d conjecture is known

to hold only in its original form: with the ordinary

(rational) Betti numbers.

To address Question 4 for Lagrangian intersec-

tions, a closely related Floer homology theory is

used. Assume L is connected, and introduce the

space of smooth paths joining L to L

ðW; L; L

¼fu : ½0; 1!W j uð0Þ2L; uð1Þ2L

Fix a point x

in L, and let u

be the path

(t) = 

). Let B be the connected component

Floer Homology 361

of (W; L, L

) containing u

.OnB we have a

symplectic action functional, defined as

f ðuÞ¼

½0;1½0;1



ð!Þ

where v : [0, 1]  [0, 1] !W is a path in B with

v(0, t) = u

(t)andv(1, t) = u(t). The symplectic

action is single valued if 

(W, L) is triv ial (even

though this condition does not guarantee that B is

simply connected). The critical points of f corre-

spond to consta nt paths whose image in W is an

intersection point of L and L

(though not all such

constant paths belong to the connected component

B). If we fix a one-parameter family of compatible

metrics g

and almost-complex structures J

on W,

then we can consider the downward gradient

trajectories of the functional. These are maps

u : R ½0; 1!W

satisfying the Cauchy–Riem ann equation

þ J



¼ 0

with boundary conditions u(s,0) 2 L and u(s,1) 2 L

With coefficients F

, a Morse complex can be

constructed much as in the case just considered. If



(W, L) is trivial, then the Floer homology group HF



obtained as the homology of this Morse complex is

isomorphic to H



(L; F

); and as a corollary, Question

4 has an affirmative answer in this case.

Without the hypothesis that 

(W, L) is trivial,

one does not expect an affirmative answer to

Question 4 in all cases. There is a ‘‘monotone’’

case, in which HF



can always be defined; but it is

not always isomorphic to H



(L; F

): instead, there is

a spectral sequence relating the two. In the general

case, there is once again the need to use rational

coefficients in place of mod 2 coefficients, in order

to deal with the orbifold nature of the trajectory

spaces that appear. This raises the question of

orientability for the trajectory spaces. In contrast

to the Morse theory for Hamiltonian diffeomorph-

isms, there is an obstruction to orie ntability,

involving spin structures on L and W. Even when

the traject ory spaces are orientable, there are further

obstructions to the existence of a Morse differential

satisfying    = 0. The theory of these obstructions

is developed in Fukaya et al. (2000). There are still

open questions in this area.

Instanton Floer Homology

A ‘‘Floer homology theory’’ for 3-manifolds should

assign to each 3-manifold Y (satisfying perhaps some

additional topological requirements) a group, say

HF(Y). Furthermore, given a four-dimensional

cobordism W from Y

to Y

, the theory should

provide a corresponding homomorphism of groups,

from HF(Y

) to HF(Y

). These homomorphisms

should satisfy the natural composition law for compo-

site cobordisms. One can formulate this by considering

the category in which an object is a closed, connected,

oriented 3-manifold Y, and in which the morphisms

from Y

to Y

are the oriented four-dimensional

cobordisms, considered up to diffeomorphism. A

Floer homology theory is then a functor from this

category (perhaps with some additional decorations or

restrictions) to the category of groups. Such a functor

was constructed by Floer (1988a), at least for the full

subcategory of homology 3-spheres (manifolds Y with

 (Y; Z) = 0). We outline the construction.

Let P !Y be a principal SU(2) bundle (necessarily

trivial). Let A denote the space of SU(2) connections

in the bundle P, and let A

be any chosen basepoint

in A. Any other A 2Acan be written as A

þ a, for

some 1-form a with values in the adjoint bundle

ad(P) whose fiber is the Lie algebra su (2). So A is an

affine space,

A¼A

þ 

ðY; adðPÞÞ

and we can identify the tangent space T

A at any

A with 

(Y;ad(P)). The Chern–Simons functional

is a smooth function

CS : A!R

depending on our choice of a reference connection

. It can be defined by stating that its derivative at

A 2Ais the linear map T

A!R given by

a 7!

trða ^ F

where F

denotes the curvature of A,asanad(P)-

valued 2-form on Y, and tr denotes the trace of a

matrix-valued 3-form. If we equip Y with a

Riemannian metric, then we have the L

inner

product on 

(Y;ad(P)), with respect to which we

can consider the gradient of CS. The formal down-

ward gradient-flow equation on A is then

ðd=dsÞA ¼F

½8

where  is the Hodge star on Y.IfA(s) is a solution

defined on an interval [s

, s

], then we can form the

corresponding four-dimensional connection A on

, s

]  Y, and eqn [8] implies that A is a solution

of the anti-self-dual Yang–Mills equation, F

= 0.

Here F

is the self-dual part of the curvature 2-form

on the cylinder. The critical points of CS are the flat

connections on Y, with F

= 0.

362 Floer Homology

Let G denote the gauge group, by which we mean

the group of automorphisms of P. When a trivializa-

tion of P is chosen, G becomes the group of smooth

maps g : Y !SU(2). A connection A 2Ais irreducible

if its stabilizer in G consists only of the constant gauge

transformations 1. The functional CS is invariant

only under the identity component of G: it descends to

afunctionCS:A=G!R=(4

Z). If we choose a

basepoint in Y, then the gauge-equivalence classes of

flat connections in A are in one-to-one correspond-

ence with conjugacy classes of representations,

 : 

ðYÞ!SUð2Þ

Given representations  and , we write M( , ) for

the quotient by G of the space of trajectories A(s)

which satisfy the gradient-flow equation [8] and

which are asymptotic to flat connections belonging

to the classes  and  as s !1. There is a purely

four-dimensional interpretation of M(, ): it can be

identified with the moduli space of solutions A to

the anti-self-dual Yang–Mills equation, or ‘‘instan-

tons,’’ on R  Y, satisfying the same asymptotic

conditions.

One defines the ‘‘instanton Floer homology’’ of Y,

roughly speaking, as the Morse homology arising

from the functional CS. In the case that Y is a

homology 3-sphere, Floer defined I



(Y) as the

homology H



(C, ) of a complex C whose generators

correspond to the irreducible representations , and

whose differential  is defined in terms of the one-

dimensional components of the moduli spaces

M(, ). To carry out the construction of I



(Y), it

is necessary to perturb the functional CS to achieve

a Morse–Smale condition: this is done by adding a

function f : A!R defined in terms of the holonomy

of connections along families of loops in Y. The

group G is not connected, and for given  and , the

moduli space M(, ) has components differing in

dimension by multiples of 8. For this reason, I



(Y)is

a Z=8-graded homology theory. It is a topological

invariant of Y, and is functorial for cobordisms, in

the manner outlined at the beginning of this section.

Various extensions have been made, to allow the

definition of I



(Y) for 3-manifolds with nontrivial H

and to incorporate the reducible representations.

Although there have been some successes (Donaldson

2002), a completely satisfactory general theory has not

been constructed. The main difficulties stem from the

noncompactness of the instanton moduli spaces (a

bubbling phenomenon) and the interaction of this

bubbling with the reducible solutions.

The instanton Floer theory for 3-manifolds is

closely tied up with Donaldson’s polynom ial invari-

ants of closed 4-manifolds, which are also defined

using the anti-self-dual Yang–Mills equations.

Seiberg–Witten Floer Homology

Seiberg–Witten Floer homology can be defined in a

manner very similar to the instanton case. Again, we

start with a Riemannian 3-manifold Y, equipped

now with a spin

structure s: a rank-2 Hermitian

vector bundle S !Y together with a Clifford multi-

plication  : 



(Y) !End(S). The configuration

space C is defined as the space of pairs (A, ),

where A is a spin

connection and  is a section of S.

In place of the Chern–Simons functional considered

above, we have the Chern–Simons–Dirac functional

CSD : C!R defined by

CSDðA; Þ¼

CS trðAÞðÞþ

h; D

id

where tr(A) denotes the connection induced by A on

the line bundle 

S and D

is the Dirac operator for

the connection A. The functional is invariant again

under the identity component of the gauge group G,

which this time is the group of maps g : Y !S

acting as automorphisms of S. The critical points are

the solutions (A, ) to the three-dimensional

‘‘Seiberg–Witten equations,’’

ðF

trðAÞ

Þð



¼ 0

 ¼ 0

in which the subscript 0 denotes the traceless part of

the endomorphism. If  and  are gauge-equivalence

classes of critical points, then we write M(, ) for

the quotient by G of the space of gradient trajec-

tories from  to .

As in the instanton case, M(, ) has a four-

dimensional interpretation: it is the quotient by the

four-dimensional gauge group of a space of solu-

tions (A, F)onR  Y to the four-dimensional

Seiberg–Witten equations:

 F

trðAÞ



ðFF



¼ 0

F ¼ 0

Here F is a section of the summand S

of the four-

dimensional spin

bundle S = S

 S



,andD

(S

) !(S



) is the four-dimensional Dirac operator.

The action of the gauge group on C is free except

at configurations with =0. These reducible con-

figurations have an S

stabilizer. Reducible critical

points of CSD correspond to flat connections in the

line bundle 

S. We can now distinguish two cases,

according to whether c

(S) is a torsion class or not.

If c

(S) is not a torsion class, then there are no flat

connections in 

S, so all critical points are

irreducible. In this case, there is a straightforward

Floer-type Morse theory for the functional CSD on

Floer Homology 363

the space C= G: for gen erators of our complex we

take the gauge-equivalence classes of critical points,

and we use the one-dimensional trajectory spaces

M(, ) to define the boundary map. The resulting

Morse homology group is denoted HM



(Y, s). It has

a canonical Z=2-grading, and is a topological

invariant of Y and its spin

structure.

If c

(S) is torsion, the theory is more complex.

There will be reducible critical points, and one

cannot exclude these from the Morse complex and

still obt ain a topological invariant of Y. One may

incorporate the reducible critical points in two

different ways, that are in a sense dual to one

another; and there is a third homology theory that

one can define, using the reducibles alone. Thus, one

can construct three Floer groups associated to Y

with the spin

structure s. The resulting theory

closely resembles the Heegaard Floer homology that

is described next.

Heegaard Floer Homology and Other

Floer Theories

Heegaard Floer homology is a Floer homology

theory for 3-manifolds that is formally similar to

Seiberg–Witten Floer homology, and conjecturally

isomorphic to it. Unlike the instanton and Seiberg–

Witten theories, its construction, due to Ozsva´th

and Szabo´, does not use gauge theory. Instead, one

begins with a decomposition of the 3-manifold into

two handlebodies with common boundary , and

one studies a symplectic manifold s

, the configu-

ration space of g-tuples of points on , where

g denotes the genus. The Heegaard Floer gro ups are

then defined by a variant of the construction used

for Lagrangian intersections (see the section ‘‘Morse

theory and the Arnol’d conjecture’’), applied to a

particular pair of Lagrangian tori in s

.

As in the case of Seiberg–Witten theory, Heegaard

Floer homology assigns to each oriented 3-manifold

Y three diff erent Floer grou ps, HF

(Y), HF



(Y), and

(Y), related by a long exact sequence:

!HF

ðYÞ!HF



ðYÞ!HF

ðYÞ!

The first two groups are dual, in that there is

a nondegenerate pairing between HF

(Y) and



(Y), where Y denotes the same 3-manifold

with opposite orientation. If W is an oriented four-

dimensional cobordism from Y

to Y

, then there

are associated functorial maps

ðWÞ : HF

ðY

Þ!HF

ðY



ðWÞ : HF



ðY

Þ!HF



ðY

ðWÞ : HF

ðY

Þ!HF

ðY

In addition, if the intersection form of W is not

negative semidefinite, there is a map

FðWÞ : HF



ðY

Þ!HF

ðY

As a special case, one can start with a closed

4-manifold X, and consider the cobordism W from

to S

obtained from X by removing two 4-balls.

In this case, the map

FðWÞ : HF



ðS

Þ!HF

ðS

encodes a diffeomorphism invariant of the original

4-manifold X. This invariant is conjectured to be

equivalent to the Seiberg–Witt en invariants of X.

Heegaard Floer homology, and its cousin Seiberg–

Witten Floer homology, have been applied success-

fully to settle long-standing problems in topology,

particularly questions related to surgery on knots.

An example of such an app lication is the theorem of

Kronheimer et al. that one cannot obtain the

projective space RP

by surgery on a nontrivial

knot in the 3-sphere.

In these and other applications of both Heegaard

and Seiberg–Witten Floer homology, two key proper-

ties of the homology groups play an important part.

The first is a nonvanishing theorem, which shows, for

example, that these Floer groups can distinguish S



from any other manifold with the same homology.

The second is a long exact sequence, which relates the

Floer groups of the manifolds obtained by three

different surgeries on a knot. The latter property is

shared by the instanton Floer groups, as was shown by

Floer (Braam and Donaldson 1995).

Other Floer-type theories have been considered,

not all of which arise from a gradient flow, but in

which the boundary map of the complex is obtained

by counting solutions to a geometric differential

equation. At the time of writing, Floer homology is

an area of very active development .

See also: Four-Manifold Invariants and Physics; Gauge

Theoretic Invariants of 4-Manifolds; Gauge Theory:

Mathematical Applications; Knot Homologies; Ljusternik–

Schnirelman Theory; Minimax Principle in the Calculus of

Variations; Moduli Spaces: An Introduction;

Seiberg–Witten Theory; Topological Quantum

Field Theory: Overview.