Simple Homotopy Types and Finite Spaces

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 ց Y of finite spaces induces a simplicial collapse K(X) ց K(Y ) of their associated simplicial complexes. Moreover, a simplicial collapse K ց L induces a collapse X (K) ց 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. Furthermore, 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.2000 Mathematics Subject Classification. 57Q10, 55U10, 57N65, 55P15.

Paths, Homotopy and Reduction in Digital Images

“…They have been used by Barmak and Minian [5] to define a collapse operation in posets which corresponds actually to the collapse operation in complexes associated to posets. The proofs of Property 8 and Theorem 6, which are out of scope of this paper, can be found in [5].…”

“…Then, the inclusion i : X \ {x} → X is a weak homotopy equivalence. [5]) Let X be a finite poset. Let x ∈ X be a simple point and K(X), K(X \ {x}) the simplicial complexes associated to X and X \ {x}.…”

Combinatorial cell complexes and Poincaré duality

“…The finite topological spaces have been studied since they were introduced by Alexandroff in [1] and the theory of simple homotopy types was developed by Whitehead in [15]. The simple homotopy types of polyhedra were studied using finite topological spaces in the recent article [3]. We refer the reader to the notes [10] and [11] for an introduction to the topology of finite spaces and to [14] for a survey of the combinatorial aspects of this theory.…”

A new spectral sequence for homology of posets

“…Examples of this interaction are simplicial homology, discrete Morse theory [8,12], the celebrated proof of Kneser's conjecture given by Lovász [10], the subsequent developments in the study of graph properties by means of topological methods [9] and the theory of finite topological spaces, which has grown considerably in the last years from works by Barmak and Minian [2,3,4,5,6].…”

“…Barmak and Minian delve deeply into this theory and obtain many interesting results, among which we mention the introduction of an elementary move in finite T 0 -spaces which corresponds exactly with the elementary collapses of simple homotopy theory of compact polyhedra [3] and a generalization of McCord's result on the weak equivalence between a compact polyhedron and its order complex [4]. Moreover, they use the theory of finite spaces to study Quillen's conjecture on the poset of non-trivial p-subgroups of a group and Andrews-Curtis' conjecture (see [2]).…”