Torsion invariant on cellular complexes

Abstract

Classifying spaces up to homeomorphism, homotopy equivalence, or combinatorial equivalence is one of the main problems in Topology. To achieve this, we have several invariants in Algebraic Topology, such as Euler characteristic, homotopy groups, homology groups and cohomology groups. In 1935, K. Reidemeister published a work on the classification of a certain class of 3-manifolds that have isomorphic homology groups and homotopy groups but are not homeomorphic. Some of them do not even have the same type of homotopy. For this classification, Reidemeister used a combinato- rial/topological invariant called torsion invariant. Based in [20], [18], [19] and [13], we will discuss here some versions of this invariant for CW-complexes, namely Whitehead torsion, Reidemeister torsion, and Reidemeister intersection torsion for pseudomanifolds with isolated singularities.