  For a Heegaard surface F in a closed orientable 3-manifold M, H(M,F) =
Diff(M)/Diff(M,F) is the space of Heegaard surfaces equivalent to the Heegaard
splitting (M,F). Its path components are the isotopy classes of Heegaard
splittings equivalent to (M,F). We describe H(M,F) in terms of Diff(M) and the
Goeritz group of (M,F). In particular, for hyperbolic M each path component is
a classifying space for the Goeritz group, and when the (Hempel) distance of
(M,F) is greater than 3, each path component of H(M,F) 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).
