In principle, Floer theory can be extended to define homotopy invariants of
families of equivalent objects (eg Hamiltonian isotopic symplectomorphisms,
–manifolds,
Legendrian knots, etc.) parametrized by a smooth manifold
. The invariant
of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence
whose term is
the homology of
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.
