Original scientific paper

An existence theorem concerning strong shape of Cartesian products

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

Abstract

The paper is devoted to the question is the Cartesian product
$X\times P$ of a compact Hausdorff space $X$ and a polyhedron $P$ a product in the strong shape category SSh of topological spaces. The question consists of two parts. The existence part, which asks whether, for a topological space $Z$, for a strong shape morphism $F\colon Z\to X$ and a homotopy class of mappings $[g]\colon Z\to P$, there exists a strong shape morphism $H\colon Z\to X\times P$, whose compositions with the canonical projections of $X\times P$ equal $F$ and $[g]$, respectively.
The uniqueness part asks if $H$ is unique. The main result of the paper asserts that $H$ exists, whenever $Z$ is either metrizable or has the homotopy type of a polyhedron. If $X$ is a metric
compactum, $H$ exists for all topological spaces $Z$. The proofs use resolutions of spaces and coherent homotopies of inverse systems. It is known that, in the ordinary shape category Sh, $H$ need not be unique, even in the case when $Z$ is a metrizable space or a polyhedron.

Keywords

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

Hrčak ID:

93191

URI

https://hrcak.srce.hr/93191

Publication date:

5.12.2012.

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