Geometry & Topology

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

Selman Akbulut

Full-text: Open access

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
Received: 4 September 2002
Accepted: 2 October 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document
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

Subjects
Primary: 57R55: Differentiable structures 57R65: Surgery and handlebodies
Secondary: 57R17: Symplectic and contact topology 57M50: Geometric structures on low-dimensional manifolds

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


Export citation

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