Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary

Abstract

In [22], we introduced absolute gradings on the three-manifold invariants developed in [21] and [20]. Coupled with the surgery long exact sequences, we obtain a number of three-and four-dimensional applications of this absolute grading including strengthenings of the "complexity bounds" derived in [20], restrictions on knots whose surgeries give rise to lens spaces, and calculations of HF + for a variety of threemanifolds. Moreover, we show how the structure of HF + constrains the exoticness of definite intersection forms for smooth four-manifolds which bound a given threemanifold. In addition to these new applications, the techniques also provide alternate proofs of Donaldson's diagonalizability theorem and the Thom conjecture for CP 2 .

Holomorphic disks and topological invariants for closed three-manifolds

“…Moreover, we fix a conformal identification (27) (observe that all the J s agree with the complex structure in this region). We can find an n-tuple of disjoint balls B ε (q i ) which are mapped into the subspace from Equation (26) by u.…”

“…where µ is the volume form for D − {q i } with the cylindrical-end metric (obtained using the cylindrical identification of Equation (28)), and the norm of ∇ (i) denotes the metric induced from the pull-back of the cylindrical metric on T Sym g (Σ − σ 1 ) (obtained from the identification (26) together with (27)). …”

Heegaard Diagrams and Holomorphic Disks

“…We have not attempted to summarize all of its properties; for additional properties, see [71], [79], [73].…”

“…(the "correction terms" of [79]) which associates to each Spin c structure the minimal Q-grading of any (non-zero) homogeneous element in HF + (Y, s) ⊗ Z Q in the image of π.…”

“…The absolute Q grading can also be calculated for lens spaces [79]. For example, for L(2, 1) ∼ = RP 3 , there are two Spin c structures s and s ′ with correction terms 1/4 and −1/4 respectively.…”