Fibrations over aspherical manifolds

Let $f\colon E \to B$ be a map between closed connected orientable manifolds. In this note, we give a necessary condition for $f$ to be a manifold fibration. In particular, we show that if $F \hookrightarrow E \stackrel{f}{\to} B$ is a fibration where $F=f^{-1}(b)$, $E$ and $B$ are closed connected triangulated orientable manifolds and $B$ is aspherical, then $f|_{E^{(n)}}\colon E^{(n)} \to B$ is surjective, where $E^{(n)}$ denotes the $n$-th skeleton of $E$ and $n=\dim B$.

Keywords

obstruction theory, fibrations, local coefficients, Shapiro’s Lemma

2010 Mathematics Subject Classification

Primary 55M20, 55R20, 55T10. Secondary 55S35.

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Published 1 January 2006