More Cappell-Shaneson spheres are standard.

*(English)*Zbl 1244.57061Summary: S. Akbulut [Ann. Math. (2) 171, No. 3, 2171–2175 (2010; Zbl 1216.57017)] 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 S. Akbulut and R. Kirby [Topology 18, 75–82 (1979; Zbl 0465.57013)] 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.

##### MSC:

57R60 | Homotopy spheres, Poincaré conjecture |

##### Keywords:

homotopy sphere; 4-manifold; Poincaré conjecture; logarithmic transformation; Gluck construction##### Software:

PARI/GP
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\textit{R. E. Gompf}, Algebr. Geom. Topol. 10, No. 3, 1665--1681 (2010; Zbl 1244.57061)

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##### References:

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