Related papers: Self-mapping Degrees of 3-Manifolds
It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in…
One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this…
We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.
Following Thurston's geometrisation picture in dimension three, we study geometric manifolds in a more general setting in arbitrary dimensions, with respect to the following problems: (i) The existence of maps of non-zero degree (domination…
We study the plane automorphisms given by polynomials with certain degree decompositions.
In this paper we explore the topological properties of self-replicating, 3-dimensional manifolds, which are modeled by idempotents in the (2+1)-cobordism category. We give a classification theorem for all such idempotents. Additionally, we…
Every closed oriented manifold $M$ is associated with a set of integers $D(M)$, the set of self-mapping degrees of $M$. In this paper we investigate whether a product $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor…
The Thurston norm of a 3-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of 3-manifolds $M$ with $b_1(M) = 2$, and whose fundamental groups admit…
We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a…
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 study the Birman exact sequence for compact $3$--manifolds, obtaining a complete picture of the relationship between the mapping class group of the manifold and the mapping class group of the submanifold obtained by deleting an interior…
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…
We describe theoretical backgrounds for a computer program that recognizes all closed orientable 3-manifolds up to complexity 8. The program can treat also not necessarily closed 3-manifolds of bigger complexities, but here some…
Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichm\"{u}ller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps.…
This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $\c{D}(M,N)$ is finite for any…
For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In…
Suppose $M$ is a closed, connected, orientable, \irr\ \3m\ such that $G=\pi_1(M)$ is infinite. One consequence of Thurston's geometrization conjecture is that the universal covering space $\widetilde{M}$ of $M$ must be \homeo\ to $\RRR$.…
The set of degrees of maps $D(M,N)$, where $M,N$ are closed oriented $n$-manifolds, always contains $0$ and the set of degrees of self-maps $D(M)$ always contains $0$ and $1$. Also, if $a,b\in D(M)$, then $ab\in D(M)$; a set…
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these…
A new lower bound on the complexity of a 3-manifold is given using the Z2-Thurston norm. This bound is shown to be sharp, and the minimal triangulations realising it are characterised using normal surfaces consisting entirely of…