The Homotopy Theory of Sphere Bundles Over Spheres (I)

James, I. M.; Whitehead, J. H. C.

doi:10.1112/plms/s3-4.1.196

Cited by 135 publications

(85 citation statements)

References 20 publications

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“…The aim of this paper is to find non-trivial examples in dimensions (4,3), (8,5) and (16,9) inspired by the early work of Antonelli ( [3,4]). The smooth maps considered in [4] are so-called Montgomery-Samelson fibrations with finitely many singularities where several fibers are pinched to points.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…There exist precisely two homotopy types among the S 3 -fibrations over S 5 which admit cross-sections (see [9], p. 217). If M 8 is a S 3 -fibration, then it should have a cross-section, because it is homotopy equivalent to S 3 × S 5 and the existence of a cross-section is a homotopy invariant (see [9], p. 196, [10], p. 164). However the two homotopy types correspond to two distinct isomorphism types as spheres bundles.…”

Section: Examples With ϕ =mentioning

confidence: 99%

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Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

Funar

Pintea

Zhang

2010

Proc. Amer. Math. Soc.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Examples With ϕ =mentioning

confidence: 99%

Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

Funar

Pintea

Zhang

2010

Proc. Amer. Math. Soc.

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“…As m − is even in the case 4B2, this implies that m − = 2, 4 or 8 by [2]. To prove our theorem, we only need to prove that (8,11) cannot occur as the multiplicities of an isoparametric hypersurface with four distinct principal curvatures. Suppose it does occur; by Theorem B the focal manifold F − is almost diffeomorphic to a S 11 -bundle over S …”

Section: Proof Of Theorem Amentioning

confidence: 99%

“…Note that m + = 11 if m − = 8 in the case 4B2. We shall prove that (8,11) cannot occur as the multiplicities and combining this with [8] it follows that Theorem A. In the case 4B2 the multiplicities (m − , m + ) of an isoparametric hypersurface with four distinct principal curvatures must be either (2,2) or (4,5).…”

Section: Introductionmentioning

confidence: 99%

On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Fang

1999

Proc. Amer. Math. Soc.

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Abstract. Let (m − , m + ) be the pair of multiplicities of an isoparametric hypersurface in the unit sphere S n+1 with four distinct principal curvatures -w.r.g., we assume that m − ≤ m + . In the present paper we prove that, in the case 4B2 of U. Abresch (Math. Ann. 264 (1983), 283-302) (i.e., where 3m − = 2(m + + 1)), m − must be either 2 or 4.As a by-product, we prove that the focal manifold F − of an isoparametric hypersurface is homeomorphic to a S

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“…Let p l : > S Q be the canonical projection. We denote P U C*Q by a <E Tr^CS*).According to [13], where p is the attaching map of the top cell of £(%). Let Y = S Q U a e n .…”

mentioning

confidence: 99%

Cohomology Classification of Self Maps of Sphere Bundles over Spheres

Morisugi¹,

Ōshima²

1996

Publ. Res. Inst. Math. Sci.

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Let % e TT^CSOCg + l)). We denote the induced g-sphere bundle over the nsphere by £(%) or simply E. The purpose of this note is to study the image of the function [£(%),£(%)] -*Hom(#*(£(%))/Tor, #*(£(%))/Tor) which assigns the induced homomorphism, where H m (X) is the reduced ra-th cohomology group of a space X with values in Z, the group of integers. Let p l : > S Q be the canonical projection. We denote P U C*Q by a 2, n > 2, and a -0 provided n = q + l.In this case, note from [5,6] where deg(a: 9 ) = q and deg(t/ n ) =n. Let kj, be integers. When #^w, a self map / of £(%) or 7 is called an M(W) -structure if /*(*<,) = /cr 9 and/*(yj = /y n . Let (fly) be a 2 X 2-matrix whose entries a^ are integers. When q =n, a self map / of E(X) or Y= S n VS n is called an (a^-) -structure with respect to {x n ,y n } if /*(£") = G,uX n +a l2 y n and /*(t/ n ) -fl2A+ G 222/n-When no confusion will occur, we will omit the words "with respect to {x n ,y n }". Notice that when q¥=n, an M(k,0-structure is an M e -structure [4] for any 6: {1,2,...} -> Z with 9 (#) =We will study conditions on the existence of an M (A:,/) -structure and an (a#) -structure on £(%) in § 2 and § 3, respectively. Our results are partial when E(%) does not have a section. To state our results, we need some notations. When E(%) has a section, we denote by f an element of Tr^CSOCg)) such that i'.(?) = X where f : S0(q) -> S0(^ + l) is the inclusion. Let j m denote the identity map of S m and /: ;r r (SO(m)) -» 7r r+m (S m ) the /-homomorphism.

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The Homotopy Theory of Sphere Bundles Over Spheres (I)

Cited by 135 publications

References 20 publications

Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$

On the topology of isoparametric hypersurfaces with four distinct principal curvatures

Cohomology Classification of Self Maps of Sphere Bundles over Spheres

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