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Using non-cofinite resolutions in shape theory. Application to Cartesian products

Sibe Mardešić ; Department of Mathematics, University of Zagreb, Zagreb, Croatia

Sažetak

The strong shape category of topological spaces SSh can be defined
using the coherent homotopy category CH, whose objects are inverse systems consisting of topological spaces, indexed by cofinite directed sets. In particular, if $X,Y$ are spaces and $\mathit{q}:Y\to \mathit{Y}$ is a cofinite HPol-resolution of $Y$, then there is a bijection between the set SSh$(X,Y)$ of strong shape morphisms $F:X\to Y$ and the set CH$(X,$ of homotopy classes $[\mathit{f}]$ of coherent homotopy
mappings $\mathit{f}:X\to \mathit{Y}$. In the paper it is shown that such a bijection exists also in the case when $\mathit{Y}$ is not cofinite. This fact makes it possible to study strong shape properties of the Cartesian product $X\times P$ of a compact Hausdorff space $X$ and a polyhedron $P$ using the standard resolution of $X\times P$, which is a non-cofinite HPol-resolution. As an application, one reduces the question
whether $X\times P$ is a product of $X$ and $P$ in the category SSh to a question concerning homotopy classes of coherent homotopy mappings.
Analogous results also hold for the ordinary shape category of topological spaces Sh and the pro-homotopy category of cofinite inverse systems of spaces.

Ključne riječi

shape; strong shape; direct product; Cartesian product; inverse limit; resolution; coherent homotopy; cofinite inverse system

Hrčak ID:

74874

URI

https://hrcak.srce.hr/74874

Datum izdavanja:

21.12.2011.

Posjeta: 1.723 *