Charged spaces (Q494968)

For a fixed path connected space \(B\) and a map \(Y\rightarrow B\), the \textit{fiberwise suspension} \(S_BY=(B\times\{-1\})\cup(Y\times[-1,1])\cup(B\times\{1\})\) is obtained by attaching the two ends of the product \(Y\times[-1,1]\) to the two copies of \(B\) by the given map \(Y\rightarrow B\) (the so called \textit{double mapping cylinder} of this map with itself). Then the composition of the two obvious maps \(B\times\{-1,1\}\rightarrow S_BY\rightarrow B\) is the projection onto the first factor. The main topic of the present paper is the \textit{Fiberwise Desuspension Problem}: For a given space \(X\) and a factorization \(B\times\{-1,1\}\rightarrow X\rightarrow B\) of the projection map, is there a space \(Y\) and a map \(Y\rightarrow B\) such that \(X=S_BY\) (up to a weak equivalence) in a way that preserves factorizations? The authors present an obstruction (in a certain metastable range) for this kind of desuspension, given in terms of a smash product in the retractive space category \(R(B)\), and they prove that the vanishing of this obstruction is both sufficient and necessary condition for the existence of a desuspension. This result is applied for obtaining a complete obstruction for compressing an embedding into \(N\times I\) to an embedding into \(N\) (where \(N\) is a compact smooth manifold).