Progress and Prospects for Floer Homology of Three Manifoldscat05/memorial/TomaszMrowka.pdfProgress and Prospects for Floer Homology of Three Manifolds Tom Mrowka MIT. What is Floer

Progress and Prospects for Floer

Homology of Three Manifolds

Tom MrowkaMIT

What is Floer Homology?

An attempt to make Morse theory for certain functionals on function spaces.

Something like a Topological Quantum Field Theory (TQFT).

Introduced in 1987 by Floer.

Morse theory

Try to get topological information about a space from dynamics of the gradient flow of a function.

The canonical example. The height function on the torus.

TQFT’s(A warm up for Floer)

Let Cob_d be the category whose objects are oriented d-dimensional manifolds and whose morphisms are cobordisms between them. Composition is given by gluing cobordisms.

A TQFT is a functor from Cob_d to the category of vector spaces and linear maps.

Axioms ofTQFT (Atiyah)

is a functor.

,

Z(I ! Y ) = IdZ(Y )

Z : Cobd ! Vect

Z(Y1

!Y2) = Z(Y1) ! Z(Y2)

Z(!Y ) = Z(Y )!

Z(!) = C

In particular a TQFT associates to each closed (d+1)-dimensional manifold a complex number.

There are lots of interesting examples for d=0,1,2. d=0 representation theory of U(n). d=1 representation theory of loop groups. d=2 is related to the Jones polynomial, the Reshetikin-Turaev-Witten invariant and Chern-Simons gauge theory.

For d=3 there are no longer such rich examples even though there are quantum field theory inspired invariants of 4-manifolds.

The Naive Idea behind TQFT.

To each d-manifold, Y, we associate a configuration space, C(Y) and construct our vector space Z(Y) by looking at some suitable space of functions on C(Y). Floer homology is the case where we would like Z(Y) to be the homology of C(Y). This is why Morse theory figures strongly in Floer homology.

W : Y1 ! Y2

To each (d+1)-manifold we associate a configuration space C(W) with maps restrictions maps:

We get maps by a pull-back push-forward construction.

A simple example would be where the C’s are oriented manifolds. Then we get a map by the composition:

W : Yin ! Yout

rin, rout : C(W ) ! C(Yin), C(Yout)

Z(W ) : Z(Yin) ! Z(Yout)

ri, ro : C ! Ci, Co

H!(Ci) ! H!(Co)

H!(Ci)PD! H

!(Ci)r!

i

! H!(C)

PD! H!(C)

(ro)!! H!(Co)

The subtleties of Floer homology

In Floer homology the configuration spaces involved are infinite dimensional.

The homology groups we would like to associate to them involve cycles that have infinite dimension and codimension.

Floer’s crucial observation is that although there no known direct rigorous construction of such a theory one can use the Morse complex to give rigorous meaning to the required homology groups.

More important problem for the TQFT viewpoint is that often the configuration spaces used have singularities. For the Seiberg-Witten equations for example the configuration space C(Y) has two strata.

It is reasonable to look for various different three manifold invariants which compute analogues of, amongst other things,

C(Y ) ! R(Y )

H!(C(Y ) \ R(Y )), andH!(C(Y ), R(Y )).

One can do this and a little more. The first instance (with very different motivation) is Ozsvath and Szabo Heegaard Floer homology. Kronheimer and M do exactly this for the Seiberg-Witten equations. The result is three functors from where the superscript means that we only allow connected non-empty three manifolds and connected cobordisms. Here is the picture.

(Ideas of Braam, Donaldson, Froyshov, Taubes are used.)

Cobo

3

These are graded modules and the action of U on these modules has degree -2.

These three functors are related by natural transformations.i,j,p

!HM, ˇHM, HM : Cobo

3 ! Z[U ] " modules

The three sphere

. . . Z 0 Z 0 Z 0 Z 0 . . .

!4 !2 0 2

. . . 0 0 0 0 Z 0 Z 0 . . .

0 2

Graded by the integers.!HM!

HM!

ˇHM∗

. . . 0 Z 0 Z 0 0 0 0 . . .

!3 !1

Short digression on 4-manifolds.

The intersection form. Given a 4-manifold possibly with boundary the cup product pairing gives rise to a non-degenerate pairing on

The rank of a maximal positive definite subspace is called

Image(!! : H2(M, "M) ! H2(M))

b+ = b

+(M)

Sieberg-Witten Invariants

In 1994 Seiberg and Witten introduced some amazing new invariants of certain closed connected and oriented 4-manifolds namely those for which >1. In their very simplest form they are simply integers assigned to every such 4-manifold.

b+

SW (M) ! Z

(Strictly speaking we need to make a further choice to take care of the sign

of the invariant.)

Can we recover the Seiberg-Witten invariant from the constructions we’ve indicated?

First idea. Take M remove two 4-balls to get a manifold X with two bounding three spheres. Try to get SW(M) from the map

!HM(X) : !HM(S3) ! !HM(S3)

(or any of the other version of Floer homology.)This fails for degree reasons. If SW(M) is notzero then necessarily this map is ZERO.

The correct method to obtain the Seiberg-Witten invariants is to use the following vanishing theorem. If then the map . In particular in this means that we have:

b+(W ) > 0

HM(W ) = 0

The diagonal map is independent of choices if . b

+(W ) > 1

Thus we can view the Seiberg-Witten invariants not as a TQFT, but rather as a “secondary TQFT”. There is a simpler TQFT whose vanishing is the obstructs the existence of the invariants.

Problem: axiomatize this situation.

HM

What can we do with Floer homology?

Taubes’ theorem says that if M admits a symplectic structure then SW(M) is non-vanishing. (1995)

Kronheimer-M generalized this to contact geometry.

Basic terms in contact geometry

A contact structure on a three manifold is a two plane field which is no-where integrable (in the sense of the Frobenius theorem). That is if the two plane field is the kernel of a one form then!

! ! d! "= 0.

Overtwisted. A contact structure is called overtwisted if there is an embedded 2-disk D so that the boundary of D is tangent to the two plane field.

Tight. A contact structure is called tight if it is not overtwisted

The basic dichotomy

ExamplesFillable. A contact structure is call fillable if Y is the boundary of a sympletic 4-manifold and the symplectic form is non-vanishing when restricted to the contact plane field.

A theorem of Eliashberg say that a fillable contact structure is tight. So the basic contact structure on the three sphere is tight

Another theorem of Eliashberg says that any orientable two plane field on a three manifold is homotopic to an overtwisted contact structure.

Invariants of contant structures (KM 1997).To each contact structure we can associate an element with the following properties.

vanishes if is overtwisted.

is non zero if is symplectically fillable.

There is an analogous invariant in Heegaard Floer theory (Oszvath-Szabo 2003)

!

!(!) ! ˇHM!("Y )

!(!)

!(!) !

!

Existence of fillable contact structures.

Theorem (Gabai, 1983) Let Y be an irreducible three manifold. Suppose S is a closed embedded surface which genus minimizing in its homology class. Then there is a taut foliation with S as a closed leaf.

Taut means that there is a closed two-form which restricts to every leaf as a volume form.

Theorem (Thurston Eliashberg 1996) Every foliation of a three manifold except the product foliation of can be perturbed to a contact structure with compatible with either orientation.

Corollary. Every irreducible three manifold which is not and has non trivial second homology carries a fillable contact structure.

S1! S

2

S1! S

2

The surgery triangle.As for ordinary homology there are exact sequences which aid in computations.

The first such sequence where established by Floer for Instanton Floer homology (we’ll see more about this theory later). (1990)

Ozsvath and Szabo proved such a sequence for Heegaard Floer theory in 2002.

They gave a second proof in 2004 which generalized to Monopole Floer theory (KMOS)

Let Y be a three manifold with torus boundary. Let be a collection of simple closed curves so that . Construct closed three manifolds by attaching a solid torus to Y so that core curve is homotopic to . Then is cobordant to by a 4-manifold . We have and .

!i, i ! Z

!i · !i+1 = !1

Yi

!i Yi

Yi+1 Wi+1/2

Wi+1/2 = Wi+7/2Yi = Yi+3

!HM(Yi+2)

!HM(Wi+5/2)

!!!!!!!!!!!!!!!!!!

!HM(Yi)dHM(Wi+1/2)

"" !HM(Yi+1)

dHM(Wi+3/2)

##""""""""""""""""

Theorem (Kronheimer-M-Ozsvath-Szabo Sep 2004).

The triangle

is exact.

ApplicationsNew proof of Culler-Gordon-Luecke-Shalen’s theorem that non-trivial surgery on a non-trivial knot cannot yield the 3-sphere.

!HM(S30(K)) ! !(!) "= 0

!!!!!!!!!!!!!!!!!!!!!!!

!HM(S3)isomorphism

"" !HM(S31(K))

##""""""""""""""""""""""

More clever applications of exact triangle yield:

Gordon’s conjecture that only surgery on the unknot can yield .

Restrictions on knots yielding lens space surgeries.

Restrictions on 3-manifolds admitting taut foliations (compare with results of Roberts, and Calegari-Dunfield.)

RP3

Eliashberg’s filling theorem.

Theorem (Eliashberg Nov 2004) Suppose Y is three manifold admitting a taut foliation. Then Y embeds in a symplectic manifold.

Corallary: The contact invariant in Heegaard Floer homology does not vanish. The KMOS results can be proved via Heegaard Floer.

Eliashberg’s result has a much more surprising corollary. Using a theorem of Feehan and Leness that a symplectic 4-manifold has non-vanishing Donaldson invariants.

Corollary (Kronheimer-M): Let Y be an irreducible three manifold which is not . Then for any cohomology class there is a representation of its fundamental group into with .

! ! H2(Y,Z2)S

1! S

2

SO3 w2(!) = "

Propetry PBing in the early 60’s while thinking about the Poincare conjecture asked to show that no counterexample could be constructed by surgery on a knot. Culler-Gordon-Luecke-Shalen reduced this to the case of +-1 surgery. A consequence of the previous corollary using the Floer’s exact triangle for instanton Floer homology is that +-1 surgery on a non-trivial knot in the three sphere yields a manifold admitting a non-trivial representation into verifying Bing.SO3

Some open questions.Develop a full version of Floer homology for Instantons (or even better for the higher rank monopole equations) that does not ignore the reducible solutions. Find the analogues of the three Floer homologies in the Seiberg-Witten case.

Axiomatize “secondary TQFTs”.

Find a rigorous framework for “semi-infinite dimensional homology”. There is important work of Manolescu in this direction.