Applications of Floer homology

Question

Can somebody tell me of other applications of Floer homology besides the proof of the Arnold conjecture. Every answer would be appreciated.

Answer 1

Floer homology has, in one form or another, become ubiquitous in symplectic geometry and to give a complete list of its applications would be a mammoth task. Here are a few.

1) One early incarnation of Floer homology was in instanton gauge theory. Floer's instanton invariant is a homology group associated to a homology 3-sphere, whose dimension 'counts' flat connections. One application of this invariant is a proof (sketched in Atiyah's paper "New invariants of 3- and 4-dimensional manifolds" section 7, and further explained in Chapter 6 of Donaldson's book "Floer homology groups in Yang-Mills theory") that a 4-manifold with nonzero Donaldson invariants (like an algebraic surface) cannot be written as a connected sum of two manifolds with $b^+>0$. There is an argument which avoids Floer homology (see Donaldson and Kronheimer section 9.3.2), but the Floer homology argument is more conceptual. The idea is that by stretching the neck around a separating 3-sphere you get a pair of 4-manifolds with boundary $S^3$ and these define for you a pair of cycles in Floer homology whose intersection product is the Donaldson invariant. Since the Floer homology of $S^3$ vanishes, so too does the Donaldson invariant. The advent of more sophisticated/computable/easily-defined invariants like Heegard-Floer or Seiberg-Witten-Floer homology has led to yet more spectacular results in low-dimensional topology.

2) In Gromov's paper on pseudoholomorphic curves, he proves that there are no embedded simply-connected Lagrangian submanifolds in a symplectic vector space. To do this he constructs a holomorphic disc with boundary on the Lagrangian using the fact that it can be displaced off itself using translation. By integrating the 1-form $\sum p_idq_i$ over the boundary of this disc you get a positive number (by Stokes's theorem and holomorphicity of the disc) which means that the boundary of the disc is nontrivial in $\pi_1$ (even $H_1$). More refined topological restrictions on Lagrangians come from more refined ways of counting pseudoholomorphic discs. Lagrangian Floer homology and its algebraic structure (better still the Fukaya category) gives such a refined framework. For instance, displaceability of a Lagrangian translates into vanishing of Floer homology and there is a spectral sequence (due to Oh) converging to Floer homology whose $E_2$ page encodes the homology of the Lagrangian. The differentials involve counts of pseudoholomorphic discs with boundary on. the Lagrangian so vanishing of Floer homology means lots of discs. The algebraic structure can give very precise control over which relative homology classes are represented by holomorphic discs. This was exploited by Buhovsky and Damian to prove monotone versions of the Audin conjecture: that (monotone) Lagrangian tori (or aspherical manifolds) in $\mathbf{C}^n$ have minimal Maslov class 2.

Of course this is only one tiny tip of the iceberg of results about Lagrangian submanifolds proved using Floer theory which include the works of Fukaya-Oh-Ohta-Ono, Biran-Cornea, Seidel and many others. Let me explain two more:

3) Seidel proved that if you take a pair of Lagrangian spheres $L_1,L_2$ in a symplectic 4-manifold, intersecting transversely at a single point, then you can take a neighbourhood $X$ of the two spheres (a plumbing of cotangent bundles) and inside $X$ you can Dehn twist $L_1$ around $L_2$ twice and you obtain a Lagrangian submanifold $L_3\subset X$ which is smoothly isotopic to $L_1$ but not isotopic through Lagrangian submanifolds. This Lagrangian knotting phenomenon was detected using Floer homology.

4) There is another Arnold conjecture, asserting that compact exact Lagrangian submanifolds in cotangent bundles are Hamiltonian isotopic to the zero section. Though this is known only for $T^*S^1$ and $T^*S^2$, there are suggestive results in this direction. The further structure afforded by the Fukaya category was used by Fukaya-Seidel-Smith to prove that projection of an exact Lagrangian to the zero section induces an isomorphism on homology. This result has since been strengthened by Abouzaid.

As I said, this is just a tiny selection of the applications known.

Answer 2

Lagrangian intersection Floer homology can be used to define topological invariants for knots and 3-manifold. A very successful example is Heegaard Floer homology: although it is now an almost completely combinatorial theory, its first definition used Floer homology.

In short the construction works like this: by more or less standard Morse theory one can encode every closed, connected and oriented 3-manifold by a Heegaard diagram, which is a triple $(\Sigma, \alpha, \beta)$, where $\Sigma$ is a genus $g$ closed surface, and $\alpha$ and $\beta$ are $g$-tuples of pairwise disjoint curves. Then one take the $g$-fold symmetric product $Sym^g(\Sigma)$ with two tori $T_{\alpha}= \alpha_1 \times \ldots \times \alpha_g$ and $T_{\beta}= \beta_1 \times \ldots \times \beta_g$. There is a symplectic form on $Sym^g(\Sigma)$ which makes $T_{\alpha}$ and $T_{\beta}$ Lagrangian submanifolds and a slightly modified version of $HF(T_{\alpha}, T_{\beta})$ is a very useful invariant of the 3-manifold described by $(\Sigma, \alpha, \beta)$.