Holomorphic triangles and invariants for smooth four-manifolds

Abstract

The aim of this article is to introduce invariants of oriented, smooth, closed four-manifolds, built using the Floer homology theories defined in [8] and [12]. This four-dimensional theory also endows the corresponding three-dimensional theories with additional structure: an absolute grading of certain of its Floer homology groups. The cornerstone of these constructions is the study of holomorphic disks in the symmetric products of Riemann surfaces.All of these properties (and various refinements) are stated precisely in Section 3. The invariants are constructed in Section 4, and the properties are verified in Sections 4-6.1.2. Absolute gradings. The principles used in construction of the cobordism invariant also allow us to define an absolute Q-lift of the relative Z-gradings of the Floer

Floer homology and singular knots

“…Indeed, recall [9] that the isomorphism induced on homology by isotopies can be thought of as induced by a chain map which counts holomorphic triangles. Moreover, the count of holomorphic triangles clearly takes the generator corresponding to (x, ǫ), to the generator corresponding to the same ǫ (respecting N-gradings).…”

Covering spaces and Q-gradings on Heegaard Floer homology

“…In [OS06], for c 1 (t) torsion, Ozsváth and Szabó used a bordism construction to lift gr to an absolute Q-grading on HF (Y, t). In other words, they found an absolute Q-grading gr on HF (Y, t) satisfying gr(ξ, η) = gr(ξ) − gr(η) for all homogeneous elements ξ, η ∈ HF (Y, t).…”

Heegaard Diagrams and Holomorphic Disks

“…Theorem 7.1 of [73]) the Q grading is uniquely characterized by the above property, together with the fact that d(S 3 ) = 0. The image of π determines a function…”

“…It is shown in [73] that when Y is an oriented rational homology three-sphere and s is a Spin c structure over Y , the relative Z grading on the Heegaard Floer homology described earlier can be lifted to an absolute Q-grading. …”

“…For fixed X and ξ ∈ HF + (Y 0 , s 0 ), we have that F + X,s (ξ) = 0 for all but finitely many s ∈ Spin c (X), c.f. Theorem 3.3 of [73], and hence there is a well-defined map…”