Embedded contact homology and Seiberg–Witten Floer cohomology III

Abstract

This is the third 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. 57R17; 57R57

A Contact Invariant in Sutured Monopole Homology

“…Combined with our work in [4], it also shows that the 'LOSS' invariant in knot Floer homology satisfies a certain functoriality with respect to Lagrangian concordance. Unsurprisingly, our strategy for proving Theorem 1.9 relies on the combined work of Taubes [44][45][46][47][48] and Colin et al [6][7][8], which shows that there is an isomorphism…”

Embedded contact homology and Seiberg–Witten Floer cohomology II

“…The operator depicted in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) can rightly be said to be the linearization of (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) at the given map c. A solution to (1-10) is deemed to be nondegenerate when the corresponding version of (1-11) has trivial kernel.…”

“…Then has an associated pair . ; / for use in (1)(2)(3)(4)(5)(6)(7)(8)(9), this the pair that appears in (1-1) and (1)(2)(3). Given and a positive integer m, use C .…”

“…With regards to (1)(2)(3)(4)(5)(6)(7)(8)(9)(10), note for future reference that the pairing between the differential of h at a given vortex c D .A;˛/ with a .1; 0/ tangent vector .x; Ã/ from (1-7) can be written in a number of ways. What follows are two:…”

Embedded contact homology and Seiberg–Witten Floer cohomology V

“…It is a consequence of [11,Proposition 6.4] and the bound by c 0 r 1=2 on kt . 0 /k H that (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) kD…”