Strong shape with compact supports
by
Ju.T. Lisica
Mathematical Analysis Department, Russian University of Peoples' Friendship, Mikluho-Maklay str. 6, 117198 Moscow, Russia

It is well-known that though usual homotopy category is defined for arbitrary topological spaces nevertheless its homotopy theory is fruitful and meaningful only for spaces with good local behavior, like the ANR's, polyhedra or CW-complexes, i.e., the main theorems (Hurewicz, Whitehead, Hopf, Smale etc.) of this homotopy theory are not valid in general even for compact metric spaces which are not homotopy type of ANR's. Thus, one needs the true homotopy category without mentioned defects. Strong shape category is one of such true homotopy categories and is a projective version of ANR-approximation of topological spaces. Here we introduce another one, dual to previous as an injective Comp-aproximation of topological spaces, so-called strong shape category with compact supports.

Direct systems as objects and coherent homotopy classes of coherent mappings as morphisms form the coherent homotopy category CH(inj-Top). The functor C':Ho(inj-Top) --> CH(inj-Top), induced by the coherence functor C:inj-Top --> CP(inj-Top), is an isomorphism of categories, where Ho(inj-Top) is a localization of the category inj-Top at the class \Sigma of level homotopy equivalences. We define the strong coshape category ScoSh(Top) as a category whose objects are all topological Hausdorff spaces and morphisms F:X --> Y are determined by triples (i, j, [f]), where i:X --> X, j:Y --> Y are inclusions of direct systems of compact subspaces of X and Y, respectively, cofinal in the systems of all compact subspaces and [f]:X --> Y is a morphism of CH(inj-Comp). Now we define the strong shape category with compact supports SSh_c(Top) in the same way as above but only [f]:X --> Y is a morphism of SSh(inj-Comp), where the last is a localization of the category inj-Comp at the class \Sigma of level strong shape equivalences of direct systems of compact spaces.

There is a natural functor S:ScoSh(Top) --> SSh_c(Top).

The main theorems mentioned above are valid in this category. Moreover, Steenrod-Sitnikov homology with compact supports and coherent cohomology of spaces are invariants of isomorphisms of this category. The last is the cohomology group of homotopy inverse limit of the inverse system formed by Massey cochain complexes of compact subspaces of topological space.

Date received: April 4, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-56.