Geodesic
More Cappell–Shaneson spheres are standard
Gompf, Robert E1
1Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA
Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1665-1681
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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.

DOI : 10.2140/agt.2010.10.1665
Keywords: homotopy sphere, $4$–manifold, Poincare Conjecture, logarithmic transformation, Gluck construction

Gompf, Robert E  1

1 Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA 
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Gompf, Robert E. More Cappell–Shaneson spheres are standard. Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1665-1681. doi: 10.2140/agt.2010.10.1665

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