UC Berkeley

Embedded contact homology (ECH) is a powerful three-manifold invariant that encodes information about the underlying contact manifold. It has been shown that embedded contact homology is isomorphic to certain versions of Seiberg-Witten Floer homology and Heegaard Floer homology. This thesis highlights the relations of ECH to other three-dimensional Floer theories. In Chapter 2, we use ideas from periodic Floer homology, a cousin of ECH, and techniques from pseudo-Anosov maps to answer a knot detection question in knot Floer homology. This is joint work with Ethan Farber and Braeden Reinoso. In Chapter 3, we show a connected sum formula for ECH by studying pseudo-holomorphic curves in the symplectization of the contact connected sum, without going through the tremendous isomorphisms to Seiberg-Witten Floer homology or Heegaard Floer homology. Our chain level description of the connected sum complex is useful for studying studying similar formulas for other contact homologies and ECH spectral invariants in the future. In Chapter 3, we give a relative version of ECH for contact three-manifolds with convex sutured boundaries. This generalizes both sutured ECH and Lipshitz's cylindrical reformulation of Heegaard Floer homology, and provides a potential framework for bordered ECH. This is joint work with Julian Chaidez, Oliver Edtmair, Yuan Yao and Ziwen Zhao.