On minimal Poincar\'{e} 44-complexes

We consider two types of minimal Poincar\'e $4$-complexes. One is defined with respect to the degree $1$-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincar\'e $4$-complexes were introduced by Hambleton, Kreck and Teichner. It is not based on an order relation. In the present paper we study existence and uniqueness

On the homotopy classification of 4-manifolds having the fundamental group of an aspherical 4-manifold

We study the homotopy type of closed connected oriented topological 4-manifolds whose fundamental groups are isomorphic to that of an aspherical 4-manifold. We give algebraic conditions for which such manifolds are homotopy equivalent to non-trivial connected sums

On the intersection forms of closed 4-manifolds

Given a closed 4-manifold M, let M^* be the simply-connected 4-manifol obtained from M by killing the fundamental group. We study the relation between the intersection forms \lambda_M and \lambda_(M^*). Finally, some topological consequences and examples are described

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

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

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

Exceptional surgeries on certain (1,1)(1,1)-knots

We determine certain exceptional surgeries on a 3--parametric family of hyperbolic 1--bridge genus one knots ((1,1)-knots, in short). In particular, we show that such knots admit two infinite series of lens space surgeries. Our work is related to a nice paper of Teragaito [16], since we represent his toroidal manifolds as 2--fold coverings of the 3-sphere branched over well--specified links

Special Classes of Closed Four–Manifolds

In this paper we present several results and state some open problems on the classification of topological and geometric structures of closed connected oriented (smooth) four–manifolds. In particular, we discuss many interesting classes of closed four-manifolds satisfying additional properties, that is, spin manifolds, manifolds with special homology (resp. homotopy), exact mani-folds, geometric manifolds, and smooth manifolds. The results, some of them due to the authors and their collaborators, are obtained by using methods and techniques from algebraic and differential topology, and homological algebra