The space of Heegaard splittings

Abstract

Abstract For a Heegaard surface Σ in a closed orientable 3-manifold M, we denote by ℋ(M, Σ) = Diff(M)/Diff(M, Σ) the space of Heegaard surfaces equivalent to the Heegaard splitting (M, Σ). Its path components are the isotopy classes of Heegaard splittings equivalent to (M, Σ). We describe H(M, Σ) in terms of Diff(M) and the Goeritz group of (M, Σ). In particular, for hyperbolic M each path component is a classifying space for the Goeritz group, and when the (Hempel) distance of (M, Σ) is greater than 3, each path component of ℋ(M, Σ) is contractible. For splittings of genus 0 or 1, we determine the complete homotopy type (modulo the Smale Conjecture for M in the cases when it is not known).