Related papers: Embedding Heegaard Decompositions

A general method for the construction of smooth flat connections on 3-manifolds is introduced. The procedure is strictly connected with the deduction of the fundamental group of a manifold M by means of a Heegaard splitting presentation of…

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…

Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The…

We study the way a strongly irreducible Heegaard surface $\Sigma$ intersects a knot exterior $X$ embedded in a 3-manifold, and show that if $\Sigma \cap \partial X$ consists of simple closed curves which are essential in both $\Sigma$ and…

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give…

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…

In this paper, we show that if a closed, connected, oriented 3-manifold M = M1#M2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on M_1 and M_2. We also give an explicit…

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that…

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…

In this paper, we give an algorithm to build all compact orientable atoroidal Haken 3-manifolds with tori boundary or closed orientable Haken 3-manifolds, so that in both cases, there are embedded closed orientable separating incompressible…

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$, then $G$…

Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such…

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $\Sigma^2M$ as wedge sums of some elementary…

We show that if the Hempel distance of a Heegaard splitting is larger than three then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that…

Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a…

Morse functions are important objects and tools in understanding topologies of manifolds since the 20th century. Their classification has been natural and difficult problems, and surprisingly, this is recently developing. Since the 2010's,…

The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The…

Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf…

Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism \phi. We show that when \phi is "sufficiently complicated" then (1) the…

Closed (and simply-connected) manifolds whose dimensions are greater than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed…