A closed simplicial model category for proper homotopy and shape theories

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" TTOE, 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.

Brown representability for exterior cohomology and cohomology with compact supports

“…For the onto character of the second map it is enough to apply [12, Prop. 4.1.21] or [16,Prop. 4.2] which, for convenience, we recall here: let…”

On Proper and Exterior Sequentiality

“…The corresponding exterior space will be denoted by X cc . The correspondence X → X cc defines a full embedding [10,Thm. 3.2]:…”

“…Therefore, several classical sequential results cannot be transferred to the proper category. A way to solve this problem is to consider a greater category having better categorical properties and then define a convenient notion of sequential object that captures the one of ω-sequential space when we restrict to P. The category of exterior spaces (see [10,11]) is a good possibility. Broadly speaking, an exterior space is a topological space with a 'neighborhood system at infinity', while an exterior map is a continuous map which is 'continuous at infinity'.…”

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

“…The category of exterior spaces has been provided with a well developed homotopy theory ( [12,[16][17][18]21]). The study of the exterior and proper homotopy invariants has proved to be useful in the study of non-compact manifolds ( [8,30]), the study of the shape of some compact spaces ( [22]), the L-S proper category ( [14,15]), et cetera.…”

“…An answer to this problem is given by the notion of exterior space. The new category of exterior spaces and maps is complete and cocomplete and contains as a full subcategory the category of spaces and proper maps, see [16,17]. We refer to [9,11,12,21,22] for further properties and applications of exterior homotopy, and to [26] for a survey of proper homotopy.…”