Approximately fibering a manifold over an aspherical one
A map \(p: M \;\rightarrow \;B\) between closed manifolds is a manifold approximate fibration (MAF) iff and only if the preimage of any open ball in \(B\) has the weak homotopy type of the homotopy fiber of \(p\). A key intermediate result of this paper is that given a continuous map \(p: M \rightarrow B\) that satisfies certain conditions there exists a homotopy equivalence \(f: M \rightarrow E\) with \(E\) a compact ENR and an approximate fibration \(q : E \rightarrow B\) with \(p = q\circ f\). Given such a factorization denote by \(\tau(f) \in Wh(\pi_{1}(E))\) the Whitehead torsion of the homotopy equivalence \(f\). The image of this in the cokernel \(NWh(p)\) of the assembly map \[ H_{1}^{\pi}(\tilde{B}, Wh(p)) \rightarrow \pi_{1}(Wh(M)) \] is called the tight torsion. A naive conjecture is that the map \(p\) is homotopic to a MAF if and only if \(N\tau(p) = 0\). The authors give an example to show that this conjecture is false. However they also give several conditions where they show that the conjecture is true. A stabilization of \(p\) is a map \(M \times T^{s} \rightarrow B \times T^{s}\), where \(T^{s}\) is the \(s\)-dimensional torus. The authors show that in case \(B\) is PL and the map \(p\) is surjective on \(\pi_{1}\), the homotopy fibre of \(p\) is finitely dominated and the \(L\)-theoretic Farrrell-Jones conjecture is true for \(\pi_{1}(M)\) then there exists an \(s\) such that the stabilized map is a MAF. The main results then follow from theorems on splitting such a stabilization.
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