A classification of S3-bundles over S4

Crowley, Diarmuid; Escher, Christine

doi:10.1016/s0926-2245(03)00012-3

Cited by 39 publications

(73 citation statements)

References 19 publications

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“…By [KiSh01] we know that the Berger space may be given an orientation so as to have the oriented P L-type of some S 3 -bundle M m,10 over S 4 with Euler class 10 and Pontrijagin class 2 (10+2m) ∈ Z ∼ = H 4 (S 4 ) with respect to the standard generator (in the notation of [CE00]). In fact, because the value of the P L-invariant s 1 (M ) = 28 ek(M ) ∈ Q/Z of [KS88] equals Given a pair of 2-connected, 7-manifolds M 1 and M 2 , they are P L-homeomorphic to each other if and only if there exists an exotic sphere Σ so that M 1 #Σ = M 2 .…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the EK-invariant is additive with respect to connected sums and attains 28 distinct values on the group of exotic 7-spheres. The previous two facts were used in [CE00] to do the diffeomorphism classification of S 3 -bundles over S 4 . Hence, M 1 and M 2 are oriented diffeomorphic if and only if they are P L-homeomorphic and have the same EK-invariant.…”

Section: Introductionmentioning

confidence: 99%

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Diffeomorphism type of the Berger space SO (5)/SO (3)

Goette¹,

Kitchloo²,

Shankar³

2004

ajm

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diffeomorphism type of the Berger space SO (5)/SO (3)

Goette¹,

Kitchloo²,

Shankar³

2004

ajm

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“…We obtain a section for η by recalling from [5] that for each v ∈ Z there are fibre homotopy equivalences …”

Section: The Structure Sets Of S 3 × S 4 and S 4 × Smentioning

confidence: 99%

“…5 We shall move along the sequence of Theorem 1.5 from left to right: by [6] or [25] the inertia group of S p × S q is trivial and thus p+q acts freely on the base point of S Diff (S p × S q ).…”

Section: Remark 312mentioning

confidence: 99%

The smooth structure set of S p × S q

Crowley

2010

Geom Dedicata

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We calculate S Diff (S p × S q ), the smooth structure set of S p × S q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general S Di f f (S 4 j−1 × S 4k ) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful mapis not in general a subgroup of the topological structure set.

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“…It is worth mentioning that nonvanishing of the Euler class implies that the homotopy type of S contains at most finitely many nondiffeomorphic S 3 -bundles over S 4 [Tam58] (cf. [CE00]). It is easy to construct manifolds admitting two nonnegatively curved metrics with pairwise nonhomeomorphic souls.…”

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confidence: 99%

Vector bundles with infinitely many souls

Belegradek

2003

Proc. Amer. Math. Soc.

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Abstract. We construct the first examples of manifolds, the simplest one being S 3 ×S 4 ×R 5 , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.According to the soul theorem of J. Cheeger and D. Gromoll [CG72], a complete open manifold of nonnegative sectional curvature is diffeomorphic to the total space of the normal bundle of a compact totally geodesic submanifold, called a soul. The soul is not unique but any two souls are mapped to each other by an ambient diffeomorphism inducing an isometry on the souls [Sha79]. In this note we show that the homeomorphism type of the soul generally depends on the metric; namely the following is true. Theorem 1. There exist infinitely many complete Riemannian metrics on S3 × S 4 × R 5 with sec ≥ 0 and pairwise nonhomeomorphic souls.The proof applies some classical techniques of geometric topology to recent examples of nonnegatively curved manifolds due to K. Grove and W. Ziller [GZ00]. As we explain below, it is much easier to produce a manifold with finitely many nonnegatively curved metrics having nonhomeomorphic souls, however the full power of [GZ00] is needed to get infinitely many such metrics.Grove and Ziller [GZ00] showed that any principal S 3 ×S 3 -bundle over S 4 admits an S 3 × S 3 -invariant metric with sec ≥ 0. By O'Neill's formula, all associated bundles admit metrics with sec ≥ 0 which gives rise to a rich class of examples, including all sphere bundles over S 4 with structure group SO(4). Note that the souls in Theorem 1 are the total spaces of S 3 -bundles over S 4 with structure group SO(3). Theorem 1 is a particular case of the following. The main topological tool used in this paper is a result of L. Siebenmann [Sie69] that generalizes the famous Masur's theorem: any tangential homotopy equivalence of closed smooth n-manifolds is homotopic to a diffeomorphism after taking the

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A classification of S3-bundles over S4

Cited by 39 publications

References 19 publications

Diffeomorphism type of the Berger space SO (5)/SO (3)

Diffeomorphism type of the Berger space SO (5)/SO (3)

The smooth structure set of S p × S q

Vector bundles with infinitely many souls

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