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2008 Floer homology of families I
Michael Hutchings
Algebr. Geom. Topol. 8(1): 435-492 (2008). DOI: 10.2140/agt.2008.8.435


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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Michael Hutchings. "Floer homology of families I." Algebr. Geom. Topol. 8 (1) 435 - 492, 2008.


Received: 8 November 2007; Accepted: 2 January 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1170.57025
MathSciNet: MR2443235
Digital Object Identifier: 10.2140/agt.2008.8.435

Primary: 57R58

Keywords: Floer homology

Rights: Copyright © 2008 Mathematical Sciences Publishers


Vol.8 • No. 1 • 2008
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