---
tag: NCKFPAGHXF
kind: concept
name: Floer homology
msc: [53D40, 57R58]
sources:
  - url: https://en.wikipedia.org/wiki/Floer_homology
same_as:
  wikidata: Q652525
  wikipedia: Floer homology
---

## Summary

Floer homology is a family of homology theories built like [[Morse homology|94BWTKKGJM]] on an infinite-dimensional space.
Its chain complex is generated by critical points of an action functional.
These are, for example, periodic orbits of a Hamiltonian or intersection points of two Lagrangian submanifolds.
Its differential counts solutions of a Cauchy–Riemann type equation that connect them.
Andreas Floer introduced it in the late 1980s to prove the Arnold conjecture on fixed points of Hamiltonian diffeomorphisms.
Versions of it are now central in symplectic topology and in the topology of three- and four-manifolds.

## Explanations

### L1 human

In Morse theory one studies a function on a space through its critical points and the gradient lines that run between them.
Counting these lines gives a homology that recovers the topology of the space.
Floer applied this idea to a functional on loops in a symplectic manifold, whose critical points are periodic orbits of a Hamiltonian system.
The gradient lines become cylinders that solve a perturbed [[pseudoholomorphic curve|N469E47V75]] equation.
Take a closed symplectic manifold with $\omega = 0$ on spheres and a nondegenerate Hamiltonian diffeomorphism of it.
The resulting homology shows that the number of fixed points is at least the sum of the Betti numbers of the manifold with coefficients in $\F_2$.