Geodesic
Cappell–Shaneson’s 4–dimensional s–cobordism
Akbulut, Selman1
1 Department of Mathematics, Michigan State University, Michigan 48824, USA
Geometry & topology, Tome 6 (2002) no. 1, pp. 425-494.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

In 1987 S Cappell and J Shaneson constructed an s–cobordism H from the quaternionic 3–manifold Q to itself, and asked whether H or any of its covers are trivial product cobordism? In this paper we study H, and in particular show that its 8–fold cover is the product cobordism from S3 to itself. We reduce the triviality of H to a question about the 3–twist spun trefoil knot in S4, and also relate this to a question about a Fintushel–Stern knot surgery.

DOI : 10.2140/gt.2002.6.425
Keywords: $s$–cobordism, quaternionic space

Akbulut, Selman 1

1 Department of Mathematics, Michigan State University, Michigan 48824, USA 
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Akbulut, Selman. Cappell–Shaneson’s 4–dimensional s–cobordism. Geometry & topology, Tome 6 (2002) no. 1, pp. 425-494. doi : 10.2140/gt.2002.6.425. http://geodesic.mathdoc.fr/articles/10.2140/gt.2002.6.425/

[1] S Akbulut, A fake cusp and a fishtail, from: "Proceedings of 6th Gökova Geometry–Topology Conference" (1999) 19

[2] S Akbulut, Scharlemann's manifold is standard, Ann. of Math. $(2)$ 149 (1999) 497

[3] S Akbulut, Variations on Fintushel–Stern knot surgery on 4–manifolds, Turkish J. Math. 26 (2002) 81

[4] S Akbulut, R Kirby, An exotic involution of $S^{4}$, Topology 18 (1979) 75

[5] S Akbulut, R Kirby, A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews–Curtis conjecture, Topology 24 (1985) 375

[6] S E Cappell, J L Shaneson, Smooth nontrivial 4–dimensional $s$–cobordisms, Bull. Amer. Math. Soc. $($N.S.$)$ 17 (1987) 141

[7] S E Cappell, J L Shaneson, Corrigendum to: “Smooth nontrivial 4–dimensional $s$–cobordisms”, Bull. Amer. Math. Soc. $($N.S.$)$ 17 (1987) 401

[8] R Fintushel, R J Stern, Knots, links, and 4–manifolds, Invent. Math. 134 (1998) 363

[9] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $(2)$ 148 (1998) 619

[10] R E Gompf, Killing the Akbulut–Kirby 4–sphere, with relevance to the Andrews–Curtis and Schoenflies problems, Topology 30 (1991) 97

[11] C H Giffen, The generalized Smith conjecture, Amer. J. Math. 88 (1966) 187

[12] C M Gordon, On the higher-dimensional Smith conjecture, Proc. London Math. Soc. $(3)$ 29 (1974) 98

[13] C M Gordon, Knots in the 4–sphere, Comment. Math. Helv. 51 (1976) 585

[14] U Meierfrankenfeld, private communications

[15] T M Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. Ser. A 23 (1977) 112

[16] E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471

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