Scharlemann's manifold is standard

Selman Akbulut

In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold $f : Q \to S^3\times S^1 \,\#\, S ^2\times S^2$, and asked whether the manifold $Q$ itself is diffeomorphic to $S^3\times S^1 \,\#\, S ^2\times S^2$. Here we answer this question affirmatively.