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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.
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