Floer homology of families I

Abstract

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 filtered chain homotopy type, which gives rise to a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. The filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory, some analysis needs to be carried out in order to turn this principle into a theorem. This paper constructs the invariant in detail for the model case of finite dimensional Morse homology, and shows that it recovers the Leray-Serre spectral sequence of a smooth fiber bundle. We also generalize from Morse homology to Novikov homology, which involves some additional subtleties.

substantially revised and expanded, with new material on filtered chain homotopy type and the generalization to Novikov homology

References in corpus (1)

• Contact homology and one parameter families of Legendrian knots

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