The equivariant pair-of-pants product in fixed point Floer cohomology

Abstract

Abstract. We use equivariant methods and product structures to derive a relation between the fixed point Floer cohomology of an exact symplectic automorphism and that of its square.

A flexible construction of equivariant Floer homology and applications

“…In this section, we discuss how to define the (Borel) equivariant Floer cohomology of (L 0 , L 1 ) without the need for equivariant transversality. Other constructions with similar invariance properties are given in [54, Section 2(a)] (written in the context of Morse theory) and [51,Section 3(b)] (in the context of fixed-point Floer homology).…”

“…There are many ways to formulate precisely what one means by Floer cohomology with this parameter space. In [51,54], the authors choose a Morse function on their parameter space and couple the∂ equation to the Morse flow equation on the space. Here, we take an approach more along the lines of simplicial sets, building a homotopy coherent functor from a category E Z/2 (see Section 3.3) to the category of chain complexes, using an intermediary category of almost complex structures.…”

Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules

“…We conclude the introduction by mentioning that some other aspects of symplectomorphisms admitting a square root have been recently studied by means of "hard" symplectic topology in [45,2].…”

Connections on equivariant Hamiltonian Floer cohomology

“…[17] for applications to Gauss-Manin connections). On the geometric side, one has the equivariant squaring map [28] (5.68) HF * (φ) Q −→ HF 2 * eq (φ 2 ). This satisfies Q(qx) = q 2 Q(x), hence its image is a subspace over (Z/2)((q 2 )).…”

“…The kernel of Γ q is a subspace of the same kind. One can speculate that the composition of (5.68) and Γ q should be zero, and then further consider the relation of such statements with localisation theorems as in [28].…”