Related papers: Trisections of 3-Manifolds

Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as…

The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that…

We show that the number of stabilizations needed to interchange the handlebodies of a Heegaard splitting of a closed 3-manifold by an isotopy is bounded below by the smaller of twice its genus or half its Hempel distance. This is a…

We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional…

Let $X$ be a bundle over $S^1$ with fiber a 3--manifold $M$ and with monodromy $\varphi$. Gay and Kirby showed that if $\varphi$ fixes a genus $g$ Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$ trisections…

We establish a correspondence between trisections of smooth, compact, oriented $4$--manifolds with connected boundary and diagrams describing these trisected $4$--manifolds. Such a diagram comes in the form of a compact, oriented surface…

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 introduce the concept of a handlebody decomposition of a 3-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable 3-manifold are stably equivalent. As an…

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 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…

Generalizing Heegaard splittings of 3-manifolds and trisections of 4-manifolds, we consider multisections of higher-dimensional smooth (or PL) closed orientable manifolds, namely decompositions into 1-handlebodies whose subcollections…

Motivated by M. Scharlemann and A. Thompson's definition of thin position of 3-manifolds, we define the width of a handle decomposition a 4-manifold and introduce the notion of thin position of a compact smooth 4-manifold. We determine all…

In this note, we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by…

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented…

For each g greater than one there is a 3-manifold with two genus g Heegaard splittings that require g stabilizations to become equivalent. Previously known examples required at most one stabilization. Control of families of Heegaard…

Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $\phi:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard…

We study trisections of 4-manifolds obtained by spinning and twist-spinning 3-manifolds, and we show that, given a (suitable) Heegaard diagram for the 3-manifold, one can perform simple local modifications to obtain a trisection diagram for…

A multisection, or $n$-section, of an $(n + 1)$-dimensional manifold is a decomposition of this manifold into $n$ $1$-handlebodies of dimension $n+1$, such that all these handlebodies intersect along a closed surface, and every…

We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian…

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $\Sigma$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this,…