engleski

The S_n-equivalence of Compacta

By reducing the Marde\v{; ; ; s}; ; ; i\'{; ; ; c}; ; ; $S$-equivalence to a finite case, i.e. to each $n\in\{; ; ; 0\}; ; ; \cup\mathbb{; ; ; N}; ; ; $\ separately, we have derived the notions of $S_{; ; ; n}; ; ; $-equivalence and $S_{; ; ; n+1}; ; ; $-domination of compacta. The $S_{; ; ; n}; ; ; $-equivalence for all $n$\ coincides with the $S$-equivalence. Further, the $S_{; ; ; n+1}; ; ; $-equivalence implies $S_{; ; ; n+1}; ; ; $-domination, and the $S_{; ; ; n+1}; ; ; $-domination implies $S_{; ; ; n}; ; ; $-equivalence. The $S_{; ; ; 0}; ; ; $-equivalence is a trivial equivalence relation, i.e. all non empty compacta are mutually $S_{; ; ; 0}; ; ; $-equivalent. It is proved that the $S_{; ; ; 1}; ; ; $-equivalence is strictly finer than the $S_{; ; ; 0}; ; ; $-equivalence, and that the $S_{; ; ; 2}; ; ; $-equivalence is strictly finer than the $S_{; ; ; 1}; ; ; $-equivalence. Thus, the $S$-equivalence is strictly finer than the $S_{; ; ; 1}; ; ; $-equivalence. Further, the $S_{; ; ; 1}; ; ; $-equivalence classifies compacta which are homotopy (shape) equivalent to ANR's up to the homotopy (shape) types. The $S_{; ; ; 2}; ; ; $-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 an $n^{; ; ; \prime}; ; ; >n$ such that the $S_{; ; ; n^{; ; ; \prime}; ; ; }; ; ; $-equivalence is strictly finer than the $S_{; ; ; n}; ; ; $-equivalence.

compactum ; ANR ; shape ; S-equivalece

nije evidentirano