The moduli space of thickenings (Q2844855)

An \(n\) thickening of a finite connected CW-complex of dimension \(k\) is a pair \((M,f)\) where \(M\) is a compact \(n\)-dimensional manifold and \(f:K \rightarrow M\) is a simple homotopy equivalence. An \((n+1)\)-thickening compresses if it is the product of an \(n\)-thickening and the unit interval. In his original 1966 paper \textit{C. T. C. Wall} [Topology 5, 73-94 (1966; Zbl 0149.20501)] showed a suspension theorem relating the \(n\)-thickenings of a complex to the \(n+1\)-thickenings. This result required connectivity conditions on \(K\) namely that it is \((2k-n+1)\)-connected, and dimensional restrictions requiring \(2n \geq 3k+3\). In this paper, Mokhtar Aouina computes the homotopy fibers of this suspension map without the connectivity assumptions on \(K\). The author shows further that these homotopy fibers can be approximated by a space of sections based on a choice of a certain stable vector bundle over the complex \(K\).