Sutured ECH is a natural invariant

We show that sutured embedded contact homology is a natural invariant of sutured contact 3 3 -manifolds which can potentially detect some of the topology of the space of contact structures on a 3 3 -manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact 3 3 -manifold in the context of Seiberg–Witten Floer homology, which enables us to complete the proof of naturality.

Heegaard Floer Homology, Degree-One Maps and Splicing Knot Complements

Suppose that K and K' are knots inside the homology spheres Y and Y', respectively. Let X = Y (K, K') be the 3-manifold obtained by splicing the complements of K and K' and Z be the three-manifold obtained by 0 surgery on K. When Y' is an L-space, we use the splicing formula of [1] to show that the rank of (X ) is bounded below by the rank of (Y ) if τ(K 2) = 0 and is bounded below by rank( (Z)) − 2 rank( (Y)) + 1 if τ(K') ≠ 0.

Naturality and Mapping Class Groups in Heegaard Floer Homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.