Gluing pseudoholomorphic curves along branched covered cylinders I

Abstract

This paper and its sequel prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves u + and u − in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit γ, the total multiplicity of the negative ends of u + at covers of γ agrees with the total multiplicity of the positive ends of u − at covers of γ. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue u + and u − to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. We establish a combinatorial formula for the signed count of such gluings. As an application, we deduce that the differential ∂ in embedded contact homology satisfies ∂ 2 = 0.This paper explains the more algebraic aspects of the story, and proves the above formulas using some analytical results from part II.

Embedded contact homology and Seiberg–Witten Floer cohomology III

“…1 limit that is gauge equivalent to c C . Each d 2 P has its corresponding version of D d as given in (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11); but now viewed as a Fredholm operator mapping H to L. Quillen [11] showed (in a somewhat different context) how such operators define a real line bundle, det.D/ ! P. If d 2 P is such that either the kernel or cokernel of D d is nontrivial, then the fiber of det.D/ at a given d 2 P has a canonical identification with^m ax .kernel.…”

“…The group C 1 .M I U.1// acts on the space of solutions to (1)(2)(3)(4)(5)(6)(7)(8) as follows: Suppose that u is a map from M to U.1/ and c D .A; / is a solution to (1)(2)(3)(4)(5)(6)(7)(8). Then uc D .A u 1 du; u / is also a solution to (1)(2)(3)(4)(5)(6)(7)(8).…”

“…Use L E to denote the latter operator. Let c D .A; / 2 Conn.E/ C 1 .M I S/ denote a nondegenerate pair as defined by (1)(2)(3)(4)(5)(6)(7)(8)(9) for the given value of r . Then there is a well defined spectral flow from L E to L c .…”

“…Part 3 Fix r 1 and a coclosed 1-form for use in (1)(2)(3)(4)(5)(6)(7)(8). The associated SeibergWitten equations on R M are equations for a map, d, from R into the space Conn.E/ C 1 .M I S/.…”

“…The 1-form that appears in (1)(2)(3)(4)(5)(6)(7)(8) is constrained to lie in a certain Banach space of smooth 1-forms. The latter is denoted by and is described briefly in Section 3.d of [13].…”