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Floer homology of families I

Michael Hutchings

Algebraic & Geometric Topology 8 (2008) 435–492

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 E2 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.

Floer homology
Mathematical Subject Classification 2000
Primary: 57R58
Received: 8 November 2007
Accepted: 2 January 2008
Published: 12 May 2008
Michael Hutchings
Mathematics Department
970 Evans Hall
University of California
Berkeley, CA 94720