On the stable classification of spin four-manifolds

We study the stable classification of closed connected oriented spin smooth 4-manifolds by using techniques of Kervaire-Milnor surgery. Then we reproduce a nice result of Kurazono and Matumoto [I. Kurazono, T. Matumoto, Hiroshima Math. J. 28, 1998] for such manifolds under the assumption that the fundamental group is finitely presentable and has vanishing second and third homology with Z_2-coefficients

Manifold Spines and Hyperbolicity Equations

We give a combinatorial representation of compact connected orientable 3-dimensional manifolds with boundary and their special spines by a class of graphs with extrastructure which are strictly related to o-graphs defined and studied in [3] and [4]. Then we describe a simple algorithm for constructing the boundary of these manifolds by using a list of 6-tuples of non-negative integers. Finally we discuss some combinatorial methods for determining the hyperbolicity equations. Examples of hyperbolic 3- manifolds of low complexity illustrate in particular cases the constructions and algorithms presented in the paper

A note on irreducible Heegaard diagrams

AbstractWe construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper.We construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper

Recognizing Euclidean Space Forms with Minimal Fundamental Tetrahedra

We completely recognize the topological structure of the ten compact euclidean space forms with special minimal tetrahedra, constructed by face pairings in nice papers of Molnár [8-9]. From these polyhedral descriptions we derive special presentations with two generators for the fundamental groups of the considered manifolds. Our proofs also show that such group presentations completely characterize the euclidean space forms among closed connected $3$-manifolds. The results have also didactical importance

On graph-theoretical invariants of combinatorial manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge--colored graphs, called {sl crystallizations}. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low-dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013

An infinite sequence of non-realizable weavings

AbstractA weaving is a number of lines drawn in the plane so that no three lines intersect at a point, and the intersections are drawn so as to show which of the two lines is above the other. For each integer n⩾4 we construct a weaving of n lines, which is not realizable as a projection of a number of lines in 3-space, all of whose subfigures are realizable as such projections

Families of group presentations related to topology

AbstractWe study some algebraic properties of a class of group presentations depending on a finite number of integer parameters. This class contains many well-known groups which are interesting from a topological point of view. We find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3-manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family

Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds

We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links