Contact monoids and Stein cobordisms

TL;DR

This paper constructs Stein cobordisms related to contact 3-manifolds from open books, revealing new algebraic structures in the mapping class group and implications for contact manifold fillability.

Contribution

It introduces a Stein cobordism construction for contact manifolds associated with boundary-fixing diffeomorphisms, linking to Heegaard Floer homology and monoids in the mapping class group.

Findings

Constructed Stein cobordisms for contact connected sums and compositions.

Connected cobordisms to algebraic structures in the mapping class group.

Derived consequences for fillability of cyclic branched covers.

Abstract

Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},\xi_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},\xi_{S,h}) # (M_{S,g},\xi_{S,g}) to (M_{S,hg},\xi_{S,hg}), for any two boundary-fixing diffeomorphisms h and g. This cobordism accounts for the comultiplication map on Heegaard Floer homology discovered in an earlier paper by the author, and it illuminates several geometrically interesting monoids in the mapping class group of S. We derive some consequences for the fillability of contact manifolds obtained as cyclic branched covers of transverse knots.