Celebratio Mathematica — Kuperberg

Krystyna Maria Trybulec Kuperberg

Works connected to Nicholas Mark Stavrakas

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K. Kuperberg: “Two Vietoris-type isomorphism theorems in Borsuk’s theory of shape, concerning the Vietoris–Cech homology and Borsuk’s fundamental groups,” Chapter 22, pp. 285–313 in Studies in topology (Charlotte, NC, 14–16 March 1974). Edited by N. M. Stavrakas and K. R. Allen. Academic Press, New York. MR 383398 Zbl 0323.55021 incollection

In the realm of Borsuk’s theory of shape, where fundamental sequences play the role of mappings (see [Borsuk 1968] for definitions), we prove two isomorphism theorems of the Vietoris-type: one for Borsuk’s fundamental groups and one for Vietoris homology groups. In order to forumlate such theorems in shape theory, one has to introduce first the notion of a fundamental sequence onto (corresponding to a mapping onto) and a condition corresponding to Vietoris’ assumption on the acyclicity of the inverse images of points. Considering the special case of fundamental sequences generated by mappings, we get in Section 2 the following theorem: If \( f \) is a mapping of a pointed compactum \( (X,x_0) \) onto a pointed compactum \( (Y,y_0) \) such that \( f^{-1}(y) \) is approximately \( q \)-connected for each \( q = 0 \), 1, \( \dots, n \) and for each \( y\in Y \), then \( f \) induces an isomorphism between the \( n \)-th fundamental groups \( \underline{\pi}{}_n(X,x_0) \) and \( \underline{\pi}{}_n(Y,y_0) \). As a similar corollary for homology, we get the classical Vietoris isomorphism theorem (see [Begle 1950] and [Vietoris 1927]). A partial generalization of Smale’s Vietoris theorem for homotopy (see [Smale 1957]) is given in Section 3 as a further application of our theorem for Borsuk’s fundamental groups.

Keith Roy Allen	Related
Nicholas Mark Stavrakas	Related