Some New Four-Manifolds

Cappell, Sylvain E.; Shaneson, Julius L.

doi:10.2307/1971056

Cited by 77 publications

(82 citation statements)

References 16 publications

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“…Shaneson [7]. It took years of work with difficult Kirby calculus computations to show by the late 1980's that even the simplest of these was diffeomorphic to S 4 (see Akbulut and Kirby [5; 6], followed by Gompf [10]).…”

Section: Introductionmentioning

confidence: 99%

More Cappell–Shaneson spheres are standard

Gompf

2010

Algebr. Geom. Topol.

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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.

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Section: Introductionmentioning

confidence: 99%

More Cappell–Shaneson spheres are standard

Gompf

2010

Algebr. Geom. Topol.

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“…Such exotic free involutions on smooth manifolds are already obtained before the appearance of gauge theory, in Cappell-Shaneson [2] and Fintushel-Stern [7]. So the Theorem here may also be viewed as an extension of these works.…”

Section: Lemma Suppose That X Is a Kähler Manifold And σ Is A Free Amentioning

confidence: 76%

Smooth structures on complex surfaces with fundamental group $\mathbf {Z}_2$

Wang

1997

Proc. Amer. Math. Soc.

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“…Suppose not. Take the covering Θ : Since N 1 does not have homotopy type of real projective n-space, the universal covering N 1 of N 1 is not a homology sphere (see [2,11,19]) so that H k ( N 1 ) = 0 for some 0 < k < n. However since the degree of R…”

Section: Necessary and Sufficient Conditions For Being Codimension-2 mentioning

confidence: 99%

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

Im¹,

Kim²

2000

Proc. Amer. Math. Soc.

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Abstract. Fibrators help detect approximate fibrations. A closed, connected n-manifold N is called a codimension-2 fibrator if each map p : M → B defined on an (n + 2)-manifold M such that all fibre p −1 (b), b ∈ B, are shape equivalent to N is an approximate fibration. The most natural objects N to study are s-Hopfian manifolds. In this note we give some necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators.

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Some New Four-Manifolds

Cited by 77 publications

References 16 publications

More Cappell–Shaneson spheres are standard

More Cappell–Shaneson spheres are standard

Smooth structures on complex surfaces with fundamental group $\mathbf {Z}_2$

Necessary and sufficient conditions for s-Hopfian manifolds to be codimension-2 fibrators

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