Related papers: Degree one maps between small 3-manifolds and Heeg…

We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only…

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the…

(d+1)-colored graphs, i.e. edge-colored graphs that are (d+1)-regular, have already been proved to be a useful representation tool for compact PL d-manifolds, thus extending the theory (known as crystallization theory) originally developed…

In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating…

We offer a new proof that two closed oriented 4-manifolds are cobordant if their signatures agree, in the spirit of Lickorish's proof that all closed oriented 3-manifolds bound 4-manifolds. Where Lickorish uses Heegaard splittings we use…

Let M be a compressionbody containing a graph T (with at least one edge) such that \boundary_+ M is parallel to the union of T and \boundary_- M. We extend methods of Hayashi and Shimokawa to classify bridge surfaces for T. The results of…

Three types of rigidity theorem for orbifold elliptic genus of level N are proved. The first type deals with the case where N is relatively prime to the orders of all isotropy groups. If the top exterior power of the tangent bundle is…

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove…

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never…

In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable…

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and…

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric…

Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) =…

A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the complex of curves on a surface and a Heegaard splitting as a pair of subcomplexes generated by the equivalent diagrams. We relate geometric and combinatorial…

Let $f\colon M\to N$ be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that $N$ is not a closed graph-manifold. Suppose that $f$ induces an epimorphism on fundamental groups. We…

In this article, we prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective…

We prove a local rigidity result for infinitesimally rigid capillary surfaces in some Riemannian $3$-manifolds with mean convex boundary. We also derive bounds on the genus, number of boundary components and area of any compact two-sided…

We show that the only closed 4-manifolds admitting genus two trisections are $S^2 \times S^2$ and connected sums of $S^1 \times S^3$, $\mathbb{CP}^2$, and $\overline{\mathbb{CP}}^2$ with two summands. Moreover, each of these manifolds…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly…