Scharlemann’s manifold is standard

In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f : Q → S3 × S1#S2 × S2, and asked the question whether or not the manifold Q itself is diffeomorphic to S3 × S1#S2 × S2. Here we answer this question affirmatively. In [Sc] Scharlemann showed that if Σ3 is the Poincaré homology 3-sphere, by surgering the 4-manifold Σ× S, along a loop in Σ× 1 ⊂ Σ× S normally generating the fundamental group of Σ, one obtains a closed smooth manifold Q and homotopy equivalence: f : Q −→ S × S#S × S which is not homotopic to a diffeomorphism (actually by taking Σ to be any Rohlin invariant 1 homology sphere one gets the same result). He then posed the question whether Q is a standard copy of S3×S1#S2×S2 (i.e. whether f is a fake self-homotopy equivalence) orQ itself is a fake copy of S3×S1#S2×S2. This question has stimulated much research during the past twenty years resulting in some partial answers. For example, in [FP] it was shown that Q is stably standard, in [Sa] it was proven that it is obtained by surgering a knotted S2 ⊂ S2×S2, and in [A4] it was shown that it is obtained by “Gluck twisting” S × S#S × S along an imbedded 2-sphere. Also by a similar construction one can obtain a fake homotopy equivalence: g : P −→ S×̃S#S × S where S3×̃S1 is the nonorientable S bundle over S. But in this case it turns out that P is not standard, i.e. P is a fake copy of S3×̃S1#S2×S2 which is also obtained by Gluck twisting a 2-sphere in S3×̃S1#S2 × S ([A2], [A3], [A4]). ∗ Partially supported by NSF grant DMS-9626204 and the research at MSRI is supported by in part by NSF grant DMS-9022140.