The induced homology and homotopy functors on the coarse shape category

Abstract

In this paper we consider some algebraic invariants of the coarse shape. We introduce functors pro * -Hn and pro * -πn relating the (pointed) coarse shape category (Sh * ⋆ ) Sh * to the category pro * -Grp. The category (Sh * ⋆ ) Sh * , which is recently constructed, is the supercategory of the (pointed) shape category (Sh⋆) Sh * , having all (pointed) topological spaces as objects. The category pro * -Grp is the supercategory of the category of pro-groups pro-Grp, both having the same object class. The functors pro * -Hn and pro * -πn extend standard functors pro-Hn and pro-πn which operate on (Sh⋆) Sh * . The full analogue of the well known Hurewicz theorem holds also in Sh * ⋆ . We proved that the pro-homology (homotopy) sequence of every pair (X, A) of topological spaces, where A is normally embedded in X, is also exact in pro * -Grp. Regarding this matter the following general result is obtained: for every category C with zero-objects and kernels, the category pro-C is also a category with zero-objects and kernels, while morphisms of pro * -C generally don't have kernels.2010 Mathematics Subject Classification. 55P55, 55Q05, 55N99.

The Hurewicz theorem in Sh◦* and Sh◦*²

“…There are several versions of this theorem, relating homotopy and homology pro-groups ( [5]), pro * -groups ( [1]) and shape groups ( [5]).…”

“…The Hurewicz theorem, a fundamental result of algebraic topology that relates homotopy and homology groups, was established also for pro-groups and shape groups ( [5]) as well as for pro * -groups ( [1]), and in [4] its version for pro-coarse shape groups was given. That enabled the authors to relate coarse shape groups and coarse shape homology groups.…”