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 PoincarĂ© 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 \((S^3 \times S^1)\# (S^2\times S^2)\).


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


Zbl 0331.57007
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