Cappell-Shaneson homotopy spheres are standard. (English) Zbl 1216.57017

In 1976, S. E. Cappell and J. L. Shaneson [Ann. Math. (2) 104, 61–72 (1976; Zbl 0345.57003)] proposed a family \(\Sigma_m\), \(m \in \mathbb Z\), of candidate manifolds to be exotic copies of \(S^4\). The example \(\Sigma_0\) alone has been the subject of intense study: in 1979, S. Akbulut and R. Kirby [Topology 18, 75–82 (1979; Zbl 0465.57013)] showed that \(\Sigma_0\) was obtained from a Gluck construction of \(S^4\), and they later produced a pleasingly symmetric handlebody picture of it [S. Akbulut and R. Kirby, ibid. 24, 375–390 (1985; Zbl 0584.57009)]. Finally, R. E. Gompf [ibid. 30, No. 1, 97–115 (1991; Zbl 0715.57016)] showed that \(\Sigma_0\) is diffeomorphic to \(S^4\).
In the paper under review, S. Akbulut completes the picture by showing that the entire family \(\Sigma_m\) is diffeomorphic to \(S^4\). The proof is a clever inductive use of Kirby Calculus to reduce to the previously known case of \(\Sigma_0\).
Subsequent work of R. E. Gompf [Algebr. Geom. Topol. 10, No. 3, 1665–1681 (2010; Zbl 1244.57061)] has shown that a larger family of Cappell-Shaneson spheres are also diffeomorphic to \(S^4\).


57R60 Homotopy spheres, Poincaré conjecture
57R65 Surgery and handlebodies
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)


homotopy sphere
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