Suspending homotopy $3$-spheres and embedding mapping cylinders in $S^{4}$

Abstract:

A property of maps between closed $3$-manifolds, implied by cellularity and implying $U{V^\infty }$, is that the mapping cylinder embed locally in ${S^4}$. It is not clear what topological properties are preserved under such maps. In the present note, we show that a closed $3$-manifold admits such a map onto ${S^3}$ if and only if its suspension is ${S^4}$.