# Handlebody construction of Stein surfaces

### Abstract

The topology of Stein surfaces and contact $3$-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained—they correspond to open handlebodies with all handles of index $\le 2$. An uncountable collection of exotic $\mathbb{R}^4$’s is shown to admit Stein structures. New invariants of contact 3-manifolds are produced including a complete (and computable) set of invariants for determining the homotopy class of a $2$-plane field on a $3$-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of oriented $3$-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in $S^3$, and in each case it is seen that “most” members of the family are the
oriented boundaries of Stein surfaces.

Robert E. Gompf