The space of Heegaard splittings (Q2838632)

For a Heegaard surface \(\Sigma\) in a closed orientable \(3\)-manifold \(M\), the space \(\mathcal{H}(M,\Sigma )\) of Heegaard splittings equivalent to \((M,\Sigma )\) is defined to be the space of left cosets Diff\((M)\)/Diff\((M,\Sigma )\), where Diff\((M,\Sigma )\) is the subgroup of Diff\((M)\) consisting of diffeomorphisms taking \(\Sigma\) to \( \Sigma \). Its path-components are the isotopy classes of Heegaard splittings equivalent to \((M,\Sigma )\). The (extended) mapping class group Mod\((M,\Sigma )\) of \((M,\Sigma )\) is the quotient group of Diff\((M,\Sigma )\) modulo the component of the identity diffeomorphism, which becomes the usual mapping class group Mod\((M )\) if \(\Sigma=\emptyset \). The kernel of the natural map Mod\((M,\Sigma )\to \) Mod\((M)\) is the Goeritz group \(G(M,\Sigma )\). In this paper the authors describe \(\mathcal{H}(M,\Sigma )\) in terms of Diff\((M)\) and \(G(M,\Sigma )\).NEWLINENEWLINEIf \(\Sigma\) has genus \(\geq 2\), the components of Diff\((\Sigma )\) are contractible and the authors prove a Theorem that \(\pi_q (\)Diff\((M))\to \pi_q (\mathcal{H}(M,\Sigma ))\) is an isomorphism for \(q\geq 2\), and \(\pi_q (\mathcal{H}(M,\Sigma ))\) is an extension of \(\pi_q (\)Diff\((M))\) by the Goeritz group for \(q=1\).NEWLINENEWLINEThis Theorem applies to all \(M\) with infinite fundamental group, except for the genus \(1\) Heegaard splitting of \(S^1{\times}S^2\). In the latter case, a description of the homotopy type of \(\mathcal{H}(S^1{\times}S^2,\Sigma )\) is given in terms of the space of \(C^{\infty}\)-maps from \(S^1\) to \(S^2\).NEWLINENEWLINEWhen \(M\) is hyperbolic, it is shown that each component of \(\mathcal{H}(M,\Sigma )\) is a classifying space for the Goeritz group and if the Hempel distance \(d(M,\Sigma )\) is greater than \(3\), then \(\mathcal{H}(M,\Sigma )\) has finitely many components, each of which is contractible.NEWLINENEWLINEFinally, the authors determine the complete homotopy types of \(\mathcal{H}(L,\Sigma_n )\), for the unique Heegaard surfaces \(\Sigma_n \) in lens spaces \(L\) (including \(S^3\)) and of \(\mathcal{H}(M,\Sigma )\), for the elliptic \(3\)-manifolds \(M\) different from \(L\). (The latter results are modulo the Smale Conjecture in the cases for \(M\) and \(L\) where it is not known).