Dubrovnik VI – Geometric Topology September 30 – October 7, 2007
	ABSTRACTS	pdf-version of this abstract

Qamil Haxhibeqiri
University of Prishtina, Kosovo

The product of shape fibrations

The notion of shape fibration for maps between metric compacta was introduced by S. Mardešić and T.B. Rushing in [4] and [5]. In [3] S. Mardešić has extented this notion to maps of arbitrary topological spaces. The author has estabilished some further properties of shape fibrations in the noncompact case (see e.g. [1], [2]).

The main result of this paper is the foollowing theorem: If $p \colon E \to B$, $p' \colon E' \to B'$ are maps of arbitrary topological spaces $E, E'$ to compact Hausdorff spaces $B, B'$, then $p\times p' \colon E\times E' \to B \times B'$ is a shape fibration if and only if $p$ and $p'$ are shape fibrations. T. Watanabe in [6] has proved that the product of maps between compact Hausdorff spaces is a shape fibration if and only if each of these maps is a shape fibration. Thus, our result can be considered as a generalization of the above mentioned Watanabe's result.

In order to obtain our main result, we have also shown the following result about resolutions of product spaces: Let $\bsq=(q_{\lambda}) \colon E \to \bsE=(E_{\lambda}, q_{\lambda \lambda'}, \Lambda)$ be a morphism of pro-Top and $\bsr=(r_{\mu}) \colon B \to \bsB=(B_{\mu},r_{\mu \mu'}, M)$ a morphism of pro-Cpt such that $\bsE$ is an $ANR$-system and $\bsB$ a compact $ANR$-system. Then $\bsq\times \bsr =(q_{\lambda}\times r_{\mu}) \colon E \times B \to \bsE\times \bsB = (E_{\lambda}\times B_{\mu}, q_{\lambda \lambda'}\times r_{\mu \mu'}, \Lambda \times M)$ is a resolution of $E\times B$ if and only if $\bsq$ and $\bsr$ are resolutions of $E$ and $B$ respectively.

REFERENCES

[1] Q. Haxhibeqiri. Shape fibrations for topological spaces, Glas. Mat. 17 (37) (1982), pp. 3811–401.
[2] Q. Haxhibeqiri. The exact sequence of a shape fibration, Glas. Mat. 18 (38) (1983), pp. 1571–177.
[3] S. Marde\v si\'{c}. Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), pp. 531–78.
[4] S. Marde\v si\'{c}, T.\,B. Rushing. Shape fibrations I, Gen. Top. Appl. 9 (1978), pp. 1931–215.
[5] S. Marde\v si\'{c}, T. B. Rushing. Shape fibrations II, Gen. Top. Appl. 9(1979), pp. 2831–298.
[6] T. Watanabe. Approximative shape theory, Mimeographed Notes, Univ. of Yamaguchi, 1982.