Volume 10, issue 3 (2010)

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More Cappell–Shaneson spheres are standard

Robert E Gompf

Algebraic & Geometric Topology 10 (2010) 1665–1681
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