Properties of pseudo-holomorphic curves in symplectisations II: Embedding controls and algebraic invariants

Embedded contact homology and Seiberg–Witten Floer cohomology I

“…Thus, it induces an isomorphism between the respective .a; J / and .y a; y J / versions of H L ech . Proof of Proposition 2.4 The fact that the isomorphism preserves degree follows from the third item in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11). The fact that it intertwines the differential follows from the fourth item in (2-11).…”

“…The signs that appear in (3)(4) are also defined using families of operators; in this case the operators that appear in (3)(4)(5)(6)(7)(8)(9). A digression is needed first to say more about how this is done (See Chapter 20 of [16]).…”

“…To state the final salient property, assume now that both c and c C are irreducible solutions to (3)(4)(5) With the digression now over, assume now that r and g obey Criteria (3)(4)(5)(6)(7)(8) and (3)(4)(5)(6)(7)(8)(9)(10). What follows explains how the signs for the differential on C SW are determined.…”

“…// is a metric on kernel.a/. Assume as well that the function g has the form e with e as in (3)(4)(5)(6)(7)(8)(9)(10)(11) as defined using a given 1-form 2 . In this case, the equations in (3-5) read…”

“…With 2 fixed, and r 1 chosen, M r henceforth denotes the space of gauge equivalence classes of solutions to (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17).…”