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Floer homology of families I
Michael Hutchings
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Algebraic & Geometric Topology 8
(2008) 435–492
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Abstract
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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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Keywords
Floer homology
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Mathematical Subject Classification
Primary: 57R58
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Publication
Received: 8 November 2007
Accepted: 2 January 2008
Published: 12 May 2008
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Authors
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