Vietoris-Begle theorem and spectra

Abstract:

The following generalization of the Vietoris-Begle Theorem is proved: Suppose ${\left \{ {{E_k}} \right \}_{k \geq 1}}$ is a CW spectrum and $f:X’ \to X$ is a closed surjective map of paracompact Hausdorff spaces such that $\operatorname {Ind} X = m < \infty$. If ${f^*}:{E^k}(x) \to {E^k}({f^{ - 1}}(x))$ is an isomorphism for all $x \in X$ and $k = {m_0},{m_0} + 1, \ldots ,{m_0} + m$, then ${f^*}:{E^n}(X) \to {E^n}(X’)$ is an isomorphism and ${f^*}:{E^{n + 1}}(X) \to {E^{n + 1}}(X’)$ is a monomorphism for $n = {m_0} + m$.