Capping off open books and the Ozsvath-Szabo contact invariant

TL;DR

This paper investigates how capping off boundary components of open books affects the Heegaard Floer contact invariant, providing new tools and results for understanding contact structures and their invariants.

Contribution

It introduces a U-equivariant map on Heegaard Floer homology for capped open books and applies it to determine support genera and compute invariants for genus one open books.

Findings

Support genera of most genus one contact structures determined.

Computed 3-dimensional invariants for contact structures with non-vanishing invariants.

Established a map relating contact invariants before and after capping off boundary components.

Abstract

If (S,h) is an open book with disconnected binding then we can form a new open book (S',h') by capping off one of the boundary components of S with a disk. We define a U-equivariant map on Heegaard Floer homology which sends c^+(S',h') to c^+(S,h), and we discuss various applications. In particular, we determine the support genera of almost all contact structures compatible with genus one, one boundary component open books. In addition, we compute the 3-dimensional invariant associated to any contact structure with non-vanishing contact invariant which is compatible with a genus one open book with periodic monodromy.