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 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 spectral sequence in detail for the model case of finite-dimensional Morse theory, and shows that it recovers the Leray-Serre spectral sequence of a smooth fiber bundle. The spectral sequence for families of symplectomorphisms will be constructed in a sequel. Floer theory is a certain kind of generalization of Morse theory, of which there are now a number of different flavors. This paper introduces a fundamental structure, namely a spectral sequence invariant of families, which exists for many versions of Floer theory. In §1 we give a general description of this spectral sequence, encapsulated in the “Main Principle” below. This principle cannot be formulated as a general theorem, because there is no precise definition of “Floer theory” that encompasses all of its diverse variants. For any particular version of Floer theory, in order to turn the principle into a theorem, one needs to slightly extend the construction of the Floer theory in question and check that the requisite analysis goes through. The rest of this paper constructs the spectral sequence in detail for the model case of finite-dimensional Morse theory, in language designed to carry over to other versions of Floer theory. The spectral sequence for families of symplectomorphisms will be constructed in the sequel [19]. ∗Partially supported by NSF grant DMS-0204681.