Related papers: On p-symmetric Heegaard splittings

The rectangle condition for a genus $g$ Heegaard splitting of a 3-manifold, defined by Casson and Gordon, provides a sufficient criterion for the Heegaard splitting to be strongly irreducible. However it is unknown whether there exists a…

Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided…

This paper has been withdrawn by the author, as the proof of Theorem 3.2 contains a flaw; subsequently, both it and Theorem 3.3 are not known to hold. The content of Section 5 has been improved and expanded upon in two separate papers. The…

Heegaard splittings stratify 3-manifolds by complexity; only $S^3$ admits a genus-zero splitting, and only $S^3$, $S^1 \times S^2$, and lens spaces $L(p,q)$ admit genus-one splittings. In dimension four, the second author and Jeffrey Meier…

We give a short proof of Scharlemann's Strong Haken Theorem for closed $3$-manifolds (and manifolds with spherical boundary). As an application, we also show that given a decomposing sphere $R$ for a $3$-manifold $M$ that splits off an $S^2…

In this paper we give a method to construct Heegaard splittings of oriented graph manifolds with orientable bases. A graph manifold is a closed $3$-manifold admitting only Seifert-fibered pieces in its Jaco-Shalen decomposition; for…

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. We classify the mapping class groups of such Heegaard splittings in the case when the surface bundle has a sufficiently complicated monodromy map.

We construct families of pairs of Heegaard splittings that must be stabilized several times to become equivalent. The first such pair differs only by their orientation. These are genus n splittings of a closed 3-manifold that must be…

We introduce a simple combinatorial method for computing all versions of the knot Floer homology of the preimage of a two-bridge knot K(p,q) inside its double-branched cover, -L(p,q). The 4-pointed genus 1 Heegaard diagram we obtain looks…

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.

We define a notion of Heegaard Floer homology for three dimensional orbifolds with arbitrary cyclic singularities, generalizing the recent work of Biji Wong where the singular locus is assumed to be connected.

We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g,b)-bridge splitting of distance greater than n with respect to the…

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable…

We describe for each postive integer $k$ a 3-manifold with Heegaard surfaces of genus $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We also show that for every positive $n$, there is a…

A gap in a paper of Rubinstein-Scharlemann is explored: new examples are found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard splittings. Properties common to all the examples in the original paper are not…

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are…

The long standing classification problem in the theory of Heegaard splittings of 3-manifolds is to exhibit for each closed 3-manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve…

We give a combinatorial proof of a theorem first proved by Souto which says the following. Let M_1 and M_2 be simple 3-manifolds with connected boundary of genus g>0. If M_1 and M_2 are glued via a complicated map, then every minimal…

Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $\Sigma(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form,…

In the closed, non-Haken, hyperbolic class of examples generated by (2p,q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio p/q and determined to be unique in all…