We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse $X\searrow Y$ of finite spaces induces a simplicial collapse $\k(X)\searrow… (More)
We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence… (More)
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of… (More)
For any finite group G, we construct a finite poset (or equivalently, a finite T0-space) X, whose group of automorphisms is isomorphic to G. If the order of the group is n and it has r generators, X has n(r + 2) points. This construction improves previous results by G. Birkhoff and M.C. Thornton. The relationship between automorphisms and homotopy types is… (More)
A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and… (More)
There is a canonical way to associate two simplicial complexes K, L to any relation R ⊂ X ×Y. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker . In this article we prove a Galois-type correspondence for relations R ⊂ X × Y when X is fixed and use these constructions to investigate… (More)
In this paper we study the 2-dimension of a finite poset from the topological point of view. We use homotopy theory of finite topological spaces and the concept of a beat point to improve the classical results on 2-dimension, giving a more complete answer to the problem of all possible 2-dimensions of an n-point poset.