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On Cappell-Shaneson 4-spheres. (English) Zbl 0783.57016

S. E. Cappell and J. L. Shaneson [Ann. Math., II. Ser. 103, 349-353 (1976; Zbl 0338.57008)] constructed a family of examples, indexed by matrices in \(SL(3,\mathbb{Z}),\) of homotopy 4-spheres. Each matrix determined a pair of homotopy 4-spheres which were distinguished by a \(Z_ 2\) choice of framing, which the author terms the ”easy” and ”hard” choices of framing. Two of these examples were shown to be double covers of homotopy \(\mathbb{R}\mathbb{P}^ 4\)s which were known to be exotic by other work of S. E. Cappell and J. L. Shaneson [ibid. 104, 61-72 (1976; Zbl 0345.57003)]. Thus it was hoped that one might find a counterexample for the smooth 4-dimensional Poincaré conjecture from this class of examples. I. R. Aitchison and J. H. Rubinstein [Contemp. Math. 35, 1-74 (1984; Zbl 0567.57015)] studied an infinite family of such pairs, and showed that in each case the easy framing yielded a standard sphere. The present article constructs handle decompositions of this infinite family of Cappell-Shaneson spheres. This handle decomposition is used to show that for one choice of framing, the standard sphere is obtained. For the other choice, a handlebody with no 3-handles, and only two 1-handles and two 2-handles is obtained. The simplest member of the family is the one studied by S. Akbulut and R. Kirby [Topology 24, 375-390 (1985; Zbl 0584.57009)] and has been shown in a separate paper by the author [ibid. 30, No. 1, 97-115 (1991; Zbl 0715.57016)] to also be standard with the hard framing.

MSC:

57R60 Homotopy spheres, Poincaré conjecture
57M99 General low-dimensional topology
57Q25 Comparison of PL-structures: classification, Hauptvermutung
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References:

[1] Aitchison, I.; Rubenstein, J., Fibered knots and involutions on homotopy spheres, (Four-Manifold Theory, Contemp. Math., 35 (1984), AMS: AMS Providence, RI), 1-74 · Zbl 0567.57015
[2] Akbulut, S.; Kirby, R., Exotic involutions of \(S^4\), Topology, 18, 75-81 (1979) · Zbl 0465.57013
[3] Akbulut, S.; Kirby, R., A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture, Topology, 24, 375-390 (1985) · Zbl 0584.57009
[4] Cappell, S.; Shaneson, J., There exist inequivalent knots with the same complement, Ann. of Math., 103, 349-353 (1976) · Zbl 0338.57008
[5] Cappell, S.; Shaneson, J., Some new four-manifolds, Ann. of Math., 104, 61-72 (1976) · Zbl 0345.57003
[7] Melvin, P., Ph.D. Thesis (1977), Berkeley
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