Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.     

Fundamental groups of manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁,W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹ → M ⁿ . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.      

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

The computational complexity of basic decision problems in 3-dimensional topology

We study the computational complexity of basic decision problems of 3-dimensional topology, such as to determine whether a triangulated 3-manifold is irreducible, prime, ∂-irreducible, or homeomorphic to a given 3-manifold M. For example, we prove that the problem to recognize whether a triangulated 3-manifold is homeomorphic to a 3-sphere, or to a 2-sphere bundle over a circle, or to a real projective 3-space, or to a handlebody of genus g, is decidable in nondeterministic polynomial time (NP) of size of the triangulation. We also show that the problem to determine whether a triangulated orientable 3-manifold is irreducible (or prime) is in PSPACE and whether it is ∂-irreducible is in coNP. The proofs improve and extend arguments of prior author’s article on the recognition problem for the 3-sphere.      

Bundles, Handcuffs, and Local Freedom

We answer a question of J. Anderson's by producing infinitely many commensurability classes of fibered hyperbolic 3-manifolds whose fundamental groups contain subgroups that are locally free and not free. These manifolds are obtained by performing 0–surgery on a collection of knots with the same properties.     

Skein related links in 3-manifolds

This paper addresses two problems in the skein theory of homotopy spheres first posed by P. Traczyk. Solutions to both problems are obtained for a large class of manifolds and, since one of the basic techniques used requires the first homology group of the ambient manifold to be torsion free, the extent to which this hypothesis is actually necessary is further explored.     

Volumes of hyperbolic Löbell 3-manifolds

In 1931 F. Löbell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing Löbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 17–23, July, 1998.This research was partially supported by GARC-KOSEF (Global Analysis Research Center of National Seoul University) and by the Russian Foundation for Basic Research under grant No. 95-01-01410.     

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 ² be the universal Weierstrass family of cubic curves over ?. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 ². Since W → 𝔸 ² is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S ³ with monodromy in SL₂ (?/N).     

Approximation by light maps and parametric Lelek maps

The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every ε>0 and a map g:In→M there exists a 0-dimensional map g^′:In→M which is ε-homotopic to g. It is shown that this class has very nice properties. For example, if Mi∈AP(ni,0), i=1,2, then M₁×M₂∈AP(n₁+n₂,0). Moreover, M∈AP(n,0) if and only if each point of M has a local base of neighborhoods U with U∈AP(n,0). Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of n-dimensional Lelek maps.