Floer homology

Summary

Floer homology is a family of homology theories built like Morse homology 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.