Other kinds of contact and Floer homology

• [1] Floer homology of families I
  Algebraic and Geometric Topology 8 (2008), 435-492. Download. Erratum.
  Abstract: In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. 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 E^2 term is the homology of B with twisted 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 spectral sequence 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.

• [2] Cylindrical contact homology for dynamically convex contact forms in three dimensions (with J. Nelson),
  Journal of Symplectic Geometry 14 (2016), 983-1012. Almost final version at arxiv:1407.2898. blog post

• [3] Symplectic capacities from positive S^1-equivariant symplectic homology (with J. Gutt),
  Algebraic and Geometric Topology 18 (2018), 3537-3600. Almost final version at arXiv:1707.06514.

• [4] Axiomatic S^1 Morse-Bott theory (with J. Nelson),
  Algebraic and Geometric Topology 20 (2020), 1641-1690. Almost final version at arXiv:1711.09996.

• [5] S^1-equivariant contact homology for hypertight contact forms (with J. Nelson),
  arXiv:1906.03457, to appear in Journal of Topology.

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