Glasnik Matematicki, Vol. 37, No.1 (2002), 147-161.


Takahisa Miyata and Tadashi Watanabe

Division of Mathematics and Informatics, Faculty of Human Development, Kobe University, 3-11 Tsurukabuto, Nada-ku, Kobe, 657-8501 Japan

Department of Mathematics and Information Sciences, Faculty of Education, Yamaguchi University, Yamaguchi, 753-8513 Japan

Abstract.   Lisica and Mardesic introduced the notion of coherent expansion of a space to develop a strong shape theory for arbitrary topological spaces. Mardesic then introduced the notion of strong ANR-expansion of a space, which is an intermediate notion between ANR-resolution and ANR-expansion, and showed that this notion can be used to define the same strong shape category. The purpose of this paper is to generalize those notions to triads of spaces and show that resolutions of triads are strong expansions of triads and that strong expansions of triads are coherent expansions of triads. Hence the strong shape theory for triads is well-defined, and all notions and results on strong expansions generalize to triads of spaces. As an invariant, strong homotopy groups for triads are defined, and the excision property with respect to strong homotopy groups and Mayer-Vietoris sequances for strong homology groups are discussed.

2000 Mathematics Subject Classification.   54C56, 55P55, 55Q07.

Key words and phrases.   Triad, strong shape, resolution, strong expansion.

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