Embedded contact homology and Seiberg–Witten Floer cohomology I

Abstract

This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. This paper describes what is involved in the construction. † Supported in part by the National Science Foundation 1. If Σ ∈ M 1 (Θ -, Θ + ), then ∪ (C,m)∈Σ C is an embedded, pseudoholomorphic subvariety.

Embedded contact homology and Seiberg–Witten Floer cohomology III

“…As noted in Section 3.c of [13], the spinor bundle S for a given Spin C structure decomposes as the orthogonal direct sum E˚EK 1 where E ! M is a complex, Hermitian line bundle, and where K is now viewed as a complex line bundle.…”

“…The latter is denoted by and is described briefly in Section 3.d of [13]. The norm on this space is called the P -norm; it bounds all of the C k norms.…”

“…This isomorphism theorem is stated formally in the first paper of this series [13]. As described in Section 4 of [13], this isomorphism is obtained using two maps: The first takes generators of the embedded contact homology chain complex to generators of the Seiberg-Witten Floer cochain complex. The second associates an instanton from the Seiberg-Witten Floer cohomology differential to each pseudoholomorphic curve from the embedded contact homology differential.…”

“…The second associates an instanton from the Seiberg-Witten Floer cohomology differential to each pseudoholomorphic curve from the embedded contact homology differential. The former map is denoted in Theorem 4.2 of [13] byˆr and the latter is denoted in Theorem 4.3 of [13] by ‰ r . The mapsˆr and ‰ r are constructed in the second paper of this series [14].…”

Recent progress on symplectic embedding problems in four dimensions

“…However one can still define Φ L and prove properties (a) and (b) by passing to Seiberg-Witten theory, using arguments from ref. 23.…”