A closed simplicial model category for proper homotopy and shape theories

  1. García-Calcines, J.M. 1
  2. García-Pinillos, M. 2
  3. Hernández-Paricio, L.J. 2
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

  2. 2 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

Journal:
Bulletin of the Australian Mathematical Society

ISSN: 0004-9727

Year of publication: 1998

Volume: 57

Issue: 2

Pages: 221-242

Type: Article

DOI: HTTP://DX.DOI.ORG/10.1017/S0004972700031610 SCOPUS: 2-s2.0-0032056227 GOOGLE SCHOLAR

More publications in: Bulletin of the Australian Mathematical Society

Abstract

In this paper, we introduce the notion of exterior space and give a full embedding of the category P of spaces and proper maps into the category E of exterior spaces. We show that the category E admits the structure of a closed simplicial model category. This technique solves the problem of using homotopy constructions available in the localised category HoE and in the "homotopy category" π0E, which can not be developed in the proper homotopy category. On the other hand, for compact metrisable spaces we have formulated sets of shape morphisms, discrete shape morphisms and strong shape morphisms in terms of sets of exterior homotopy classes and for the case of finite covering dimension in terms of homomorphism sets in the localised category. As applications, we give a new version of the Whitehead Theorem for proper homotopy and an exact sequence that generalises Quigley's exact sequence and contains the shape version of Edwards-Hastings' Comparison Theorem.