Remarks to the Vietoris theorem
The author considers extensions of the classical theorem of \textit{L. Vietoris} [Math. Ann. 97, 454-472 (1927; JFM 53.0552.01)] giving homology conditions on the fibres of a surjective map \(f:X\rightarrow Y\) which insure that \(f\) induces an isomorphism in homology. A survey of the literature is given and some homotopy and cohomotopy versions of Vietoris' theorem are established e.g. conditions on a space \(P\) and map \(f:X\rightarrow Y\) are given insuring that \(f^*:[Y,P]\rightarrow [X,P]\) is bijective. The proofs are by classical obstruction theory.
scientific article; zbMATH DE number 1127706