Floer homology of families I

Abstract

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (eg 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 local coefficients in the Floer homology of the fibers. This 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.