## Scharlemann’s manifold is standard.(English)Zbl 0931.57016

In the paper [Duke Math. J. 43, 33-40 (1976; Zbl 0331.57007)], M. Scharlemann constructed a closed smooth 4-manifold $$Q$$ by surgery of the product $$\Sigma\times S^1$$, $$\Sigma$$ the Poincaré homology 3-sphere, along a loop in $$\Sigma\times 1\subset\Sigma\times S^1$$ normally generating the fundamental group of $$\Sigma$$.
Moreover, he constructed a homotopy equivalence $f: Q\to (S^3 \times S^1)\# (S^2\times S^2),$ which is not homotopic to a diffeomorphism, and asked the question whether or not $$Q$$ is diffeomorphic to $$(S^3\times S^1)\#(S^2\times S^2)$$.
This question has stimulated much research during the past twenty years resulting in some partial answers (see the references of the paper). In the present paper, the author gives a nice proof of the above open question so $$Q$$ is diffeomorphic to the connected sum $$(S^3 \times S^1)\# (S^2\times S^2)$$.

### MSC:

 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)

Zbl 0331.57007
Full Text: