Vietoris-Begle theorem
type: Item
instance of: scholarly article
title: Vietoris-Begle theorem
zbMATH Open document ID: 0571.55006
DOI: 10.1007/BF01139948
author: B. George
published in: Mathematical Notes
publication date: 1984-01-01T00:00:00Z
review text: Let \(H_*\) denote the Steenrod-Sitnikov homology, \(f: X\to Y\) a surjective map of compacta and A a countable abelian group such that \(\tilde H_ q(f^{-1}y,A)=0\) for any \(y\in Y\) and \(0\leq q<n\). The author proves that \(H_ q(f,A): H_ q(X,A)\to H_ q(Y,A)\) is an isomorphism for \(0\leq q<n\) and an epimorphism for \(q=n\). It generalizes the author's and \textit{A. Yu. Volovikov}'s result [Vestn. Mosk. Univ., Ser. I, 70-71 (1984; see the preceding review)] proved for the integer coefficients and \textit{G. E. Bredon}'s result for \(n=\infty\) [Sheaf theory (1967; Zbl 0158.205)], cf. also \textit{E. G. Begle} [Ann. Math., II. Ser. 51, 534-543 (1950; Zbl 0036.388) and Mich. Math. J. 3, No.2, 179-180 (1956; Zbl 0075.320)]. For other results on the Vietoris-Begle theorem cf. \textit{D. G. Bourgin} and \textit{R. M. Nehs} [Trans. Am. Math. Soc. 284, 281-318 (1984; Zbl 0558.55005)].
Mathematics Subject Classification ID: 55N07
Mathematics Subject Classification ID: 55N05
Mathematics Subject Classification ID: 55T99
Mathematics Subject Classification ID: 55N25
zbMATH DE Number: 3911286
zbMATH Keywords: nerve of covering
zbMATH Keywords: Mayer-Vietoris sequence
zbMATH Keywords: Steenrod-Sitnikov homology
zbMATH Keywords: surjective map of compacta
zbMATH Keywords: Vietoris-Begle theorem
reviewed by: Marek Golasiński
MaRDI profile type: Publication
cites work: The Vietoris mapping theorem for bicompact spaces
cites work: The Vietoris mapping theorem for bicompact spaces. II
cites work: Spaces of the same n-type, for all n
description: scientific article; zbMATH DE number 3911286