Trisections and link surgeries

We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby \cite{G-K3}. These links lie on Heegaard surfaces in $\#^j S^1 \times S^2$ and have surgeries yielding $\#^k S^1 \times S^2$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer.

A topologically minimal canonical Heegaard splitting of a surface bundle is critical

Every surface bundle with genus [Formula: see text] fiber has a canonical Heegaard splitting of genus [Formula: see text]. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2.

The space of Heegaard splittings

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).

CRITICAL HEEGAARD SURFACES OBTAINED BY SELF-AMALGAMATION

Critical surfaces can be regarded as topological index 2 minimal surfaces which was introduced by David Bachman. In this paper, we give a sufficient condition and a necessary condition for self-amalgamated Heegaard surfaces to be critical.

LIFTED HEEGAARD SURFACES AND VIRTUALLY HAKEN MANIFOLDS

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133–150].