Simple homotopy types and finite spaces

Jonathan Ariel Barmak,Elias Gabriel Minian

doi:10.1016/j.aim.2007.11.019

Abstract

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 ↘ Y of finite spaces induces a simplicial collapse K ( X ) ↘ K ( Y ) of their associated simplicial complexes. Moreover, a simplicial collapse K ↘ L induces a collapse X ( K ) ↘ X ( L ) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.

Highlights

There is a strong relationship between finite spaces and finite simplicial complexes, which was discovered by McCord [9]
Given a finite simplicial complex K, one can associate to K a finite T0-space X (K) which corresponds to the poset of simplices of K ordered by inclusion
One can associate to a finite T0-space X a simplicial complex K(X), whose simplices are the non-empty chains of X, and a weak homotopy equivalence K(X) → X

Summary

Introduction

One can associate to a finite T0-space X a simplicial complex K(X), whose simplices are the non-empty chains of X, and a weak homotopy equivalence K(X) → X. We wanted to find an elementary move in the setting of finite spaces (if it existed) which corresponds exactly to a simplicial collapse of the associated polyhedra We discovered this elementary move when we were looking for a homotopically trivial finite space (i.e. weak equivalent to a point) which was non-contractible. In the last section of this article we investigate the class of maps between finite spaces which induce simple homotopy equivalences between their associated simplicial complexes. To this end, we introduce the notion of a distinguished map. Φ is a simple homotopy equivalence if and only if X (φ) is a simple equivalence

Preliminaries

Simple homotopy types

Simple homotopy equivalences