## Geometry & Topology

### Cappell–Shaneson's $4$–dimensional $s$–cobordism

Selman Akbulut

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 6, Number 1 (2002), 425-494.

Dates
Accepted: 2 October 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882802

Digital Object Identifier
doi:10.2140/gt.2002.6.425

Mathematical Reviews number (MathSciNet)
MR1943756

Zentralblatt MATH identifier
1021.57014

Keywords
$s$–cobordism quaternionic space

#### Citation

Akbulut, Selman. Cappell–Shaneson's $4$–dimensional $s$–cobordism. Geom. Topol. 6 (2002), no. 1, 425--494. doi:10.2140/gt.2002.6.425. https://projecteuclid.org/euclid.gt/1513882802

#### References

• S Akbulut, A fake cusp and a fishtail, Turkish Journal of Math. 23 (1999) 19–31.
• S Akbulut, Scharlemann's manifold is standard, Ann. of Math. 149 (1999) 497–510.
• S Akbulut, Variations on Fintushel–Stern knot surgery on $4$–manifolds, Proceedings of 8th Gokova Geometry–Topology Conf. (2001) 81–92.
• S Akbulut, R Kirby, An exotic involution of $S^{4}$, Topology, 18 (1979) 75–81
• S Akbulut, R Kirby, A potential smooth counterexample in dimension $4$ to the Poincare conjecture, the Andrews–Curtis conjecture, Topology, 24 (1985) 375–390
• S E Cappell, J L Shaneson, Smooth nontrivial $4$–dimensional $s$–cobordisms, Bull. Amer. Math.Soc. 17 (1987) 141–143.
• S E Cappell, J L Shaneson, Corrigendum to: Smooth nontrivial $4$–dimensional $s$–cobordisms, Bull. Amer. Math. Soc. 17 (1987) 401
• R Fintushel, R Stern, Knots, links, and $4$–manifolds, Inv. Math. 134 (1998) 363–400.
• R Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619–693
• R Gompf, Killing the Akbulut–Kirby $4$–sphere, with relevance to the Andrews–Curtis and Schoenflies problems, Topology, 30 (1991) 97–115
• C H Giffen, The Generalized Smith Conjecture, Amer. J. Math. 88 (1966) 187–198
• C M Gordon, On the Higher-Dimensional Smith Conjecture, Proc. London Math. Soc. 29 (1974) 98–110
• C M Gordon, Knots in the 4–sphere, Comm. Math. Helv. 51 (1976) 585–596
• U Meierfrankenfeld, (Private communications)
• T Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. 23 (1977) 112–128
• E C Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965) 471–495