Freedman, Michael There is no room to spare in four-dimensional space. (English) Zbl 0538.57001 Notices Am. Math. Soc. 31, 3-6 (1984). 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 Cited in 5 Documents 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 Keywords:exotic real 4-spaces; quadratic forms of 4-manifolds; smooth structures on 4-space; Whitney trick PDFBibTeX XMLCite \textit{M. Freedman}, Notices Am. Math. Soc. 31, 3--6 (1984; Zbl 0538.57001)