Floer homology of families

Morse homology recovers the homology of a manifold from the gradient flow lines of a Morse function on it. Floer theory is a generalization of Morse theory from finite dimensional to certain infinite dimensional settings. In finite dimensions one can recover classical topological invariants other than just homology from gradient flow lines of Morse functions, and in principle these invariants have Floer theoretic analogues which might also be interesting. For example, a straightforward variant of standard constructions recovers the Leray-Serre spectral sequence of a family of manifolds (i.e. a smooth fiber bundle) from flow lines associated to a family of (generically) Morse functions on them. The goal of this project is to develop analogues of this construction in various kinds of Floer theory, to obtain possibly interesting invariants of families of other kinds of objects.

The simplest case of Floer theoretic invariants of families is for families parametrized by S^1. Here the ``continuation map'' around S^1 defines a ``monodromy'' automorphism of the Floer homology of the basepoint of the family. A version of this monodromy was introduced by Seidel in the 1990's for loops of Hamiltonian symplectomorphisms. My former student Tamas Kalman used this idea in his thesis to show that the inclusion of the space of Legendrian knots in R^3 into the space of smooth knots induces a noninjective map on the fundamental group. Frederic Bourgeois has independently used a related construction to detect some nontrivial families of contact structures parametrized by S^k.

[1] Floer homology of families I
Algebraic and Geometric Topology 8 (2008), 435-492. download.
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. 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 spectral sequence 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.
In preparation (slowly, because I have been focused on embedded contact homology): Floer homology of families II (or something like that).
This paper will discuss Floer-theoretic invariants of families of symplectomorphisms, following [1].

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