Simple homotopy types and finite spaces

JONATHAN ARIEL BARMAK

Arolla, Suiza

Conferencia; Third Arolla Conference on Algebraic Topology; 2008

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We introduce the notion of a collapse of finite spaces  and show that a collapse from X to Y induces a simplicial collapse between the associated (order) complexes K(X) and K(Y). Conversely, a simplicial collapse induces a collapse between the associated finite spaces (=face posets). Therefore the functor K establishes a one-to-one correspondence between simple homotopy types (of finite CW-complexes) and simple equivalence classes of finite spaces. Finite spaces seem to be very appropriate to study classical simple homotopy theory because of their combinatorial and topological nature and since the elementary collapses in this setting are very simple to handle and describe: they consist of removing a single point of the space.