The S_n-equivalence of compacta

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\u27s 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\u27 > n such that the Sn\u27-equivalence is strictly finer than the Sn-equivalence