Floer Homology - Floer Homology of Three-manifolds

Floer Homology of Three-manifolds

There are several conjecturally equivalent Floer homologies associated to closed three-manifolds. Each yields three types of homology groups, which fit into an exact triangle. A knot in a three-manifold induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal properties to the combinatorially-defined Khovanov homology.)

These homologies are closely related to the Donaldson and Seiberg invariants of 4-manifolds, as well as to Taubes's Gromov invariant of symplectic 4-manifolds; the differentials of the corresponding three-manifold homologies to these theories are studied by considering solutions to the relevant differential equations (Yang–Mills, Seiberg–Witten, and Cauchy–Riemann, respectively) on the 3-manifold cross R. The 3-manifold Floer homologies should also be the targets of relative invariants for four-manifolds with boundary, related by gluing constructions to the invariants of a closed 4-manifold obtained by gluing together bounded 3-manifolds along their boundaries. (This is closely related to the notion of a topological quantum field theory.) For Heegaard Floer homology, the 3-manifold homology was defined first, and an invariant for closed 4-manifolds was later defined in terms of it.

There are also extensions of the 3-manifold homologies to 3-manifolds with boundary: sutured Floer homology (Juhász 2008) and bordered Floer homology (Lipshitz, Ozsváth & Thurston 2008). These are related to the invariants for closed 3-manifolds by gluing formulas for the Floer homology of a 3-manifold described as the union along the boundary of two 3-manifolds with boundary.

The three-manifold Floer homologies also come equipped with a distinguished element of the homology if the three-manifold is equipped with a contact structure beginning with Kronheimer and Mrowka in the Seiberg–Witten case. (A choice of contact structure is required to define embedded contact homology but not the others.　For embedded contact homology see Hutchings (2009))

These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by assigning homotopy classes of 2-plane fields) for SWF, and using the SWF-ECH isomorphism, ECH.