Vector Fields on Spheres

Adams, J. F.

doi:10.2307/1970213

Cited by 634 publications

(459 citation statements)

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“…In [31] it is shown that H*(F(2k), Z/p) is isomorphic as a Hopf algebra to (g) E{aan^\0<j<6n}®An/ie-An n prime to p n+fc=Omod(p- 1) where 6n is the smallest integer t such that Xnp< is p-divisible by exactly p* and £ is the Frobenius map x -* xp.…”

Section: 'Pkoxh •••])mentioning

confidence: 99%

“…To simplify the description, suppose also that no is a unit in Zip\ (i.e., A(d+i)/2 is p-divisible by exactly p) and that d > 2. 1.2. THEOREM.…”

Section: 'Pkoxh •••])mentioning

confidence: 99%

“…The image of J. For any fc > 0 Adams [1] has defined an B-map ipk: BU -► BU which, in the p-local setting, is an infinite loop map if fc is prime to p. These maps arise from the if-theory operations…”

Section: Examplesmentioning

confidence: 99%

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Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Lance¹

1988

Trans. Amer. Math. Soc.

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ABSTRACT. When localized at an odd prime p, the classifying space PL/O for smoothing theory splits as an infinite loop space into the product CxN where C = Cokernel (J) and N is the fiber of a p-local H-map BU -> BU. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their fc-invariants to properties of the defining self-maps of BU. If Y is a smooth manifold, the set of homotopy classes [Y, N] is a certain subgroup of resmoothings of Y, and the ^-invariants of N generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of Y and frequently vanish. Introduction.When localized at an odd prime p, the classifying spaces for smoothings and for surgery theory can be written as products involving the p-local factors BO, C (the cokernel of J), and the fibers of certain B-maps BU -► BU. This paper studies the latter spaces, computing the cohomology of their Postnikov towers and relating the fc-invariants to the underlying geometry they classify. In smoothing theory there is an infinite loop space decomposition PL/O ss G x N where N is the fiber of a self-map of BU whose homotopy is the p-torsion in the groups of homotopy spheres bounding parallelizable manifolds. We show that for many smooth manifolds Y the resmoothings classified by N can be effectively computed and given a "characteristic variety" sort of description.Any B-map /: BU -► BU is determined up to homotopy by the characteristic sequence A = (Ai,A2,...), where /» is multiplication by Xj on Ti2j(BU) = Ztpy Because of the infinite loop space decomposition BU ss W x Q2W x ■■ ■ x U2p~4W [4,46] where n^W) = Zip) in dimensions t = 2j(p -1), we may write / as a product of B-maps with corresponding factorization of the fiber

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Section: 'Pkoxh •••])mentioning

confidence: 99%

“…To simplify the description, suppose also that no is a unit in Zip\ (i.e., A(d+i)/2 is p-divisible by exactly p) and that d > 2. 1.2. THEOREM.…”

Section: 'Pkoxh •••])mentioning

confidence: 99%

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Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Lance¹

1988

Trans. Amer. Math. Soc.

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“…It was proved by Radon [5], Hurwitz [2] and Adams [1] that a" = span S"~x can be obtained as the maximal number of linear transformations of R" which induce linearly independent vector fields on S"~x <L-* R".…”

mentioning

confidence: 99%

“…We recall that (see [1], [2], [5] Let Xx,. .. ,XS be linear vector fields on Gk(R") and J*!.7^ the corresponding skew-symmetric transformations of 7?"…”

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confidence: 99%

Linear vector fields on 𝐺̃_{𝑘}(𝐑ⁿ)

Leite¹,

Miatello²

1980

Proc. Amer. Math. Soc.

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Abstract. We determine the maximal number of linearly independent vector fields on the grassmannian of oriented £-subspaces of R", which are induced by linear transformations of Ak(R").Introduction. It was proved by Radon [5], Hurwitz [2] and Adams [1] that a" = span S"~x can be obtained as the maximal number of linear transformations of R" which induce linearly independent vector fields on S"~x span S"~x and in §4 we provide examples where equality or strict inequality occurs.

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