More Cappell–Shaneson spheres are standard

Gompf, Robert E

doi:10.2140/agt.2010.10.1665

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More Cappell–Shaneson spheres are standard

Robert E Gompf

Algebraic & Geometric Topology 10 (2010) 1665–1681

DOI: 10.2140/agt.2010.10.1665

Abstract

Akbulut has recently shown that an infinite family of Cappell–Shaneson homotopy $4$ –spheres is diffeomorphic to the standard $4$ –sphere. In the present paper, a different method shows that a strictly larger family is standard. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akbulut and Kirby showing that the simplest example with untwisted framing is standard. Instead, hidden symmetries of the original Cappell–Shaneson construction are exploited. In the course of the proof, an example is given showing that Gluck twists can sometimes be undone using symmetries of fishtail neighborhoods.

Keywords

homotopy sphere, $4$–manifold, Poincare Conjecture, logarithmic transformation, Gluck construction

Mathematical Subject Classification 2000

Primary: 57R60

References

Publication

Received: 5 March 2010

Revised: 5 June 2010

Accepted: 8 June 2010

Published: 11 August 2010

Authors

Robert E Gompf
Department of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX 78712-0257 USA

Volume 10, issue 3 (2010)