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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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[1] S. Akbulut, ”Scharlemann’s manifold is standard,” Ann. of Math., vol. 149, iss. 2, pp. 497-510, 1999. · Zbl 0931.57016
[2] S. Akbulut, ”Cappell-Shaneson’s 4-dimensional \(s\)-cobordism,” Geom. Topol., vol. 6, pp. 425-494, 2002. · Zbl 1021.57014
[3] S. Akbulut, ”The Doglachev surface,” 2008.
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[5] S. Akbulut and R. Kirby, ”An exotic involution of \(S^4\),” Topology, vol. 18, iss. 1, pp. 75-81, 1979. · Zbl 0465.57013
[6] S. Akbulut and R. Kirby, ”A potential smooth counterexample in dimension \(4\) to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture,” Topology, vol. 24, iss. 4, pp. 375-390, 1985. · Zbl 0584.57009
[7] S. E. Cappell and J. L. Shaneson, ”Some new four-manifolds,” Ann. of Math., vol. 104, iss. 1, pp. 61-72, 1976. · Zbl 0345.57003
[8] R. E. Gompf, ”Killing the Akbulut-Kirby \(4\)-sphere, with relevance to the Andrews-Curtis and Schoenflies problems,” Topology, vol. 30, iss. 1, pp. 97-115, 1991. · Zbl 0715.57016
[9] R. E. Gompf, ”On Cappell-Shaneson \(4\)-spheres,” Topology Appl., vol. 38, iss. 2, pp. 123-136, 1991. · Zbl 0783.57016
[10] M. Freedman, R. E. Gompf, S. Morrison, and K. Walker, ”Man and machine thinking about the smooth 4-dimensional Poincare conjecture,” 2009. · Zbl 1236.57043
[11] R. Kirby, ”A calculus for framed links in \(S^3\),” Invent. Math., vol. 45, iss. 1, pp. 35-56, 1978. · Zbl 0377.55001
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