An addendum to the Vietoris-Begle theorem
A version of the classical Vietoris-Begle theorem involving Alexander- Spanier (A-S) cohomology with coefficients in a group G asserts: If \(f: X\to Y\) is a continuous closed surjective map between compact metrizable spaces such that \(H^*(f^{-1}(y);G)=0\) for \(*<n\) where \(n\geq 0\), then the induced map \(H^*(f)\) of the cohomology groups is: (1) an isomorphism when \(*<n\), and (2) a monomorphism when \(*=n\). Under these hypotheses, instead of conclusions (1) and (2), the following strengthened conclusion is proved: The sequence \[ 0 \to \tilde H^*(Y;G) \to \tilde H^*(X;G) \to \prod_{y\in Y}\tilde H^*(f^{- 1}(y);G) \] of reduced A-S cohomology groups is exact, where the middle and last homomorphisms are induced by f and inclusions. A dual Vietoris- Begle theorem by using integral A-S homology groups was recently proved by Volovikov and Nguen (a reference is given). A proof of this theorem is derived by using the techniques of this paper. Analogous results for Steenrod homology and pro-homology are also given. These results are too technical to be stated here. These results comprise a useful ''addendum'' to the classical Vietoris-Begle theorem.
scientific article; zbMATH DE number 3964969
scientific article