Floer homology of families I

By Michael Hutchings. 2003

49 citations recorded by the source corpus.

Floer theory is a certain kind of generalization of Morse theory, of which there are now a number of different flavors. These give invariants of symplectomorphisms, 3‐manifolds, Legendrian knots, and many other types of objects. This paper describes a fundamental structure which apparently exists in most or all versions of Floer theory. The structure in question is a homotopy invariant of families of equivalent objects parametrized by a smooth manifold B . Its different manifestations thus give invariants of families of Hamiltonian isotopic symplectomorphisms, families of 3‐manifolds, etc. 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 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. The general properties of this family invariant are stated in the “Main Principle” below. This principle cannot be formulated as a general theorem, because there is no precise definition of “Floer theory” that encompasses all of its diverse variants. For any particular version of Floer theory, in order to turn the principle into a theorem, one needs to slightly extend the construction of the Floer theory in question and check that the requisite analysis goes through. The rest of this paper constructs the invariant in detail for the model version of Floer theory, namely finite-dimensional Morse homology,

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b50e4e96efd3ea0735c28b161958fb27f218d533

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. P. Seidel
2. I. Smith
3. A. Oancea
4. Frederick J. Bourgeois III
5. K. Fukaya
6. M. Abouzaid
7. Alexander F. Ritter
8. Ben Webster

In Geometry and Topology

1. P. Seidel
2. M. Abouzaid
3. I. Smith
4. A. Oancea
5. Frederick J. Bourgeois III
6. Alexander F. Ritter
7. T. Ekholm
8. Mark McLean

In Mathematical Physics

1. P. Seidel
2. I. Smith
3. M. Abouzaid
4. A. Oancea
5. Frederick J. Bourgeois III
6. Alexander F. Ritter
7. T. Ekholm
8. D. Auroux

Citations in the corpus

• The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint (2007, methodology): “discussing a new approach using family Floer cohomology [10] and the “wrapped Fukaya category”.”
• $S^1$-equivariant symplectic homology and linearized contact homology (2012, background): “The second one is inspired by Seidel’s [55, §8b], based on an idea of Hutchings [42], see also Seidel and Smith [57, §2.”
• Localization for Involutions in Floer Cohomology (2010, methodology): “This is the approach used to define general family Floer homology theories in [ H ]; a related idea for S 1 -equivariant Floer homology appears”