Related papers: Trisections of 3-Manifolds

For a 3-manifold $M$ and an acyclic $\mathit{SL}(2,\mathbb{C})$-representation $\rho$ of its fundamental group, the $\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion $\tau_\rho(M) \in \mathbb{C}^\times$ is defined. If there are only finitely…

Let $M=W\cup_T V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a…

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A…

We study complete $3$-manifolds with nonnegative scalar curvature under additional regularity assumptions. We prove that a contractible such manifold is diffeomorphic to $\mathbb{R}^3$, and that an open handlebody admitting such a metric…

We show that if $V^3$ is a handlebody in $\R^3$, with curves $J_1, ..., J_g \subset \partial V$ which are the attaching curves for a Heegaard splitting of a homology sphere, then there exists a homeomorphism $h\colon V \to V$ so that each…

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…

We consider a Heegaard splitting M=H_1 \cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty…

We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R…

Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link,…

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped…

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both…

We classify a large class of "unbalanced" 4-manifold GK-trisections, which are a slight generalization of 4-manifold trisections defined by Gay and Kirby.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, which is called a real structure. A real contact 3-manifold is a real 3-manifold with a contact distribution that is antisymmetric with…

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a…

A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In…

We show that the problem of deciding whether a closed three-manifold admits an elliptic structure lies in NP. Furthermore, determining the homeomorphism type of an elliptic manifold lies in the complexity class FNP. These are both…

We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…

According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been…

The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this…

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show…