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There is no room to spare in four-dimensional space. (English) Zbl 0538.57001

This article is a semi-popular exposition of the ideas leading to the recent discovery of a manifold V that is homeomorphic to Euclidean 4- space \({\mathbb{R}}^ 4\) but which is not diffeomorphic to \({\mathbb{R}}^ 4\). It depends on looking at the intersection of 2k-submanifolds of a 4k- manifold \(V^{4k}\), and the author shows that when pairs of such intersections cancel algebraically, then they are candidates for geometric cancellation provided there is ’room’ in the 4k-manifold to apply the ’Whitney trick’ (the punch-line of which is irritatingly omitted). From this he leads on to an explanation of a recent theorem of Donaldson, that if the intersection matrix of a smooth, simply connected \(V^ 4\) is positive definite then it is integrally equivalent to \(\pm I\). Next, he constructs a submanifold V of \(S^ 2\times S^ 2\) which by h-cobordism is homeomorphic to \({\mathbb{R}}^ 4\), and then shows that if V were diffeomorphic to \({\mathbb{R}}^ 4\) then by adding it to other 4- manifolds as a connected sum Donaldson’s theorem would be violated.
Reviewer: H.B.Griffiths

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R55 Differentiable structures in differential topology
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