## 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 $$S^4$$.

### MSC:

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

homotopy sphere
Full Text:

### References:

 [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. [4] I. R. Aitchison and J. H. Rubinstein, ”Fibered knots and involutions on homotopy spheres,” in Four-Manifold Theory, Providence, RI: Amer. Math. Soc., 1984, vol. 35, pp. 1-74. · Zbl 0567.57015 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.