An exotic 4-manifold. (English) Zbl 0839.57016

Summary: In [ibid., 335-356 (1991; Zbl 0839.57015), see the review above] we have constructed a fake smooth structure on a contractible 4-manifold \(W^4\) relative to boundary. This is a smooth manifold \(V\) with \(\partial V = \partial W\) such that the identity map \(\partial V \to \partial W\) extends to a homeomorphism but not to a diffeomorphism \(V \to W\). This is a relative result in the sense that \(V\) itself is diffeomorphic to \(W\), even though no such diffeomorphism can extend the identity map on the boundary. Here we strengthen this result by dropping the boundary hypothesis at the expense of slightly enlarging \(W\): We construct two compact smooth 4-manifolds \(Q_1\), \(Q_2\) which are homeomorphic but not diffeomorphic to each other. In particular no diffeomorphism \(\partial Q_1 \to \partial Q_2\) can extend to a diffeomorphism \(Q_1 \to Q_2\).


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R55 Differentiable structures in differential topology
57R80 \(h\)- and \(s\)-cobordism


Zbl 0839.57015
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