Gauged Hamiltonian Floer homology I: Definition of the Floer homology groups

Floer homology of families I

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a spectral sequence whose E2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. For any parti...

Hamiltonian handleslides for Heegaard Floer homology

A g-tuple of disjoint, linearly independent circles in a Riemann surface Σ of genus g determines a ‘Heegaard torus’ in its g-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides a new route to the handleslide-invarianceof ...

Floer homology groups in Yang–Mills theory

Fukaya - Floer Homology

We determine the Fukaya-Floer (co)homology groups of the threemanifold Y = Σ × S, where Σ is a Riemann surface of genus g ≥ 1. These are of two kinds. For the 1-cycle S ⊂ Y , we compute the Fukaya-Floer cohomology HFF ∗(Y, S) and its ring structure, which is a sort of deformation of the Floer cohomology HF ∗(Y ). On the other hand, for 1-cycles δ ⊂ Σ ⊂ Y , we determine the Fukaya-Floer cohomolo...

Floer Homology and Invariants of Homology Cobordism

By using surgery techniques, we compute Floer homology for certain classes of integral homology 3-spheres homology cobordant to zero. We prove that Floer homology is two-periodic for all these manifolds. Based on this fact, we introduce a new integer valued invariant of integral homology 3-spheres. Our computations suggest its homology cobordism invariance.