Original scientific paper

https://doi.org/10.3336/gm.42.1.14

The S_n-equivalence of compacta

Nikica Uglešić ; University of Zadar, Studentski dom, F. Tuđmana 24 D, 23000 Zadar, Croatia
Branko Červar ; Department of Mathematics, University of Split, Teslina ulica 12/III, 21000 Split, Croatia

Abstract

By reducing the Mardešiæ S-equivalence to a finite case, i.e., to each n {0} N separately, we have derived the notions of Sn-equivalence and Sn+1-domination of compacta. The Sn-equivalence for all n coincides with the S-equivalence. Further, the Sn+1-equivalence implies Sn+1-domination, and the Sn+1-domination implies Sn-equivalence. The S0-equivalence is a trivial equivalence relation, i.e., all non empty compacta are mutually S0-equivalent. It is proved that the S1-equivalence is strictly finer than the S0-equivalence, and that the S2-equivalence is strictly finer than the S1-equivalence. Thus, the S-equivalence is strictly finer than the S1-equivalence. Further, the S1-equivalence classifies compacta which are homotopy (shape) equivalent to ANR's up to the homotopy (shape) types. The S2-equivalence class of an FANR coincides with its S-equivalence class as well as with its shape type class. Finally, it is conjectured that, for every n, there exists n' > n such that the Sn'-equivalence is strictly finer than the Sn-equivalence.

Keywords

Compactum; ANR; shape; S-equivalence

Hrčak ID:

12894

URI

https://hrcak.srce.hr/12894

Publication date:

12.6.2007.

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