Vietoris-Begle theorems for nonclosed maps (Q367299)

The well-known Vietoris-Begle theorem states that every Vietoris map (i.e., acyclic closed surjective map) \(f: X\to Y\) induces an isomorphism \(H^q(f): H^q(Y) \to H^q(X)\) for each \(q\in{\mathbb Z}\). Here \(H^{\ast}\) denotes the Alexander-Spanier cohomology functor with coefficients in an arbitrary abelian group. In this paper the author considers a class \({\mathcal V}\) of maps which contains all Vietoris maps and some non closed and non acyclic maps. More precisely, the class \({\mathcal V}\) consists of maps \(f: X\to Y\) between paracompact spaces such that there is an isomorphism NEWLINE\[NEWLINE\varinjlim \{H^q(f^{-1}(V)) \mid \text{\(V\) is an open neighborhood of \(y\)}\} \approx H^q(pt)NEWLINE\]NEWLINE for each \(y\in Y\) and \(q\in{\mathbb Z}\). The main result of the paper is that the Vietoris-Begle theorem holds for the class \({\mathcal V}\). The author also considers a more general setting. For any subsets \(G, C\subset X\) with \(G\) being closed, a map \(f: X\to Y\) is a Vietoris map with respect to \((G, C)\) if \(f(G\cap C) = Y\) and there is an isomorphism NEWLINE\[NEWLINE\varinjlim \{H^q(f^{-1}(W\cap C)) \mid \text{\(W\) is an open neighborhood of \(f^{-1}(y) \cap G\)}\} \approx H^q(pt)NEWLINE\]NEWLINE for each \(y\in Y\) and \(q\in{\mathbb Z}\). The theorem states that every Vietoris map \(f: X\to Y\) with respect to \((G, C)\) such that \(f|_G\) is a closed map induces an isomorphism NEWLINE\[NEWLINEH^q(Y) \to \varinjlim \{H^q(W\cap C) \mid \text{\(W\) is an open neighborhood of \(G\)} \}NEWLINE\]NEWLINE for each \(q\in{\mathbb Z}\). A fibrewise version of the Vietoris-Begle theorem is also discussed.