Fibrations over aspherical manifolds (Q2488616)

Let \(p:E\to B\) be a manifold fibration with typical fibre \(F\). Assume that \(F\), \(E\), and \(B\) are 0-connected closed triangulated orientable manifolds, \(n:=\dim B\) and \(\dim E=n+k\). Denote by \(E^{(n)}\) the \(n\)-skeleton of \(E\). The authors prove that \(p| \,E^{(n)}:E^{(n)}\to B\) is surjective provided \(\pi_q(B)=0\) for \(2\leq q\leq k+1\). Moreover, for \(n\geq2\) the primary obstruction \(o_n(p,\bar{b})\in H^n(E;\mathbb{Z}\pi_B^*)\) is a non-zero integer where \(\bar{b}\) is the constant map at \(b\in B\).