On Symplectic Caps

Abstract

An important class of contact 3-manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3-manifolds. As an application, we present a new obstruction for such singularities to admit rational homology disk smoothings.We dedicate this paper to Oleg Viro on the occasion of his 60th birthday.

Weighted homogeneous singularities and rational homology disk smoothings

“…No example of a normal surface singularity with nonstar-shaped minimal good resolution graph which admits a QHD smoothing is known. Partial results regarding the nonexistence of QHD smoothings follow from [4,16,18], but the lack of a convenient and general compactifying divisor prevents us from treating the general case with methods similar to the ones applied in the present paper.…”

Floer homology, twist coefficients, and capping off

“…where Y 0 is obtained from Y by Dehn surgery along B 1 with framing given by that of the pages of the open book decomposition, and W 0 is the trace of the surgery. The cobordism W 0 often admits a natural symplectic structure ( [GS12], [Wen13]), and hence provides a natural operation between the contact structures ξ (S,φ) and ξ (S 0 ,φ 0 ) . Nonetheless, the invariants {τ B i (S, φ)} and {τ B i (S 0 , φ 0 )} (which provide valuable information regarding ξ (S,φ) and ξ (S 0 ,φ 0 ) , respectively) often differ greatly.…”

Capping off open books and the Ozsváth–Szabó contact invariant

“…If φ is periodic with non-negative fractional Dehn twist coefficients, then the same is true of φ ′ , and it follows from Proposition 6.3 that the contact invariants c + (S g,r−1 , φ ′ ) and c + (S g,r , φ) have well-defined Q-gradings. Since F + W,s 0 sends c + (S g,r−1 , φ ′ ) to c + (S g,r , φ), by Theorem 1.2, the grading shift formula in [28] gives (14) gr(c + (S g,r , φ)) − gr(c + (S g,r−1 , φ ′ )) = c 1 (s 0 ) 2 − 2χ(W ) − 3σ(W ) 4 .…”

“…One expects that a similar construction should produce a symplectic structure on the cobordism W considered in Theorem 1.2. In fact, since this paper first appeared, Gay and Stipsicz have shown that one can find a symplectic form on W for which the contact 3-manifolds supported by (S g,r , φ) and (S g,r−1 , φ ′ ) are strongly concave and strongly convex, respectively [14]. On the other hand, the contact invariant in Heegaard Floer homology (in contrast with its analogue in Monopole Floer homology [20]) is not known, in general, to behave naturally with respect to the map induced by a strong symplectic cobordism, and so Theorem 1.2 provides new information in this regard.…”