Sequential homology

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a 'system of open neighborhoods at infinity'. An exterior map is a continuous map which is 'continuous at infinity'. A strongly locally finite CW-complex X, whose skeletons are provided with the family of the complements of compact subsets, can be considered as an exterior space X. Associated with a compact metric space we also consider the open fundamental complex OFC(X); introduced by Lefschetz. In this paper we use sequences of cycles converging to infinity to introduce 'ordinary' sequential homology and cohomology theories in the category of exterior spaces. One of the interesting differences with respect to the ordinary theories of topological spaces is that the role of a point is played by the exterior space N of natural numbers with the discrete topology and the cofinite externology. For a strongly locally finite CW-complex X, we see that the singular homology of X is isomorphic to Hseq(X; 0 ), the locally finite homology is isomorphic to Hseq(X; 0 ) and the end homology is isomorphic to Hseq(X; 0 /0 cohomology one has that the compact support cohomology is isomorphic to Hseq(X; 0 ), the singular cohomology is isomorphic to Hseq(X; 0 ) and the end cohomology is isomorphic to Hseq(X; 0). With respect to the Lefschetz fundamental complex, one has that the ech homology of a compact metric space can be found as a subgroup of Hseq(OFC(X); ); the Steenrod homology is isomorphic to H+1seq (OFC(X);; 0 /0 ) and the ech cohomology of X is isomorphic to Hseq(OFC(X);; 0 /0 ). Finally, one also has a Poincaré isomorphism Hseqq(M) Hn-qseq (M), where M is a triangulable, second countable, orientable, n-manifold. We remark that in both sides of the isomorphism we are using sequential theories. © 2001 Elsevier Science B.V. All rights reserved

Homotopia propia simplicial

Centro de Informacion y Documentacion Cientifica (CINDOC). C/Joaquin Costa, 22. 28002 Madrid. SPAIN / CINDOC - Centro de Informaciòn y Documentaciòn CientìficaSIGLEESSpai

A closed simplicial model category for proper homotopy and shape theories

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" 0E, 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