Computing v1-periodic Homotopy Groups of Spheres and some Compact Lie Groups

Topology and its Applications

2008

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Section: Remark 33mentioning

confidence: 98%

Homotopy type and v1-periodic homotopy groups of p-compact groups

Topology and its Applications

2008

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“…In [8], the first author and Mahowald defined the ( p-primary) v 1 -periodic homotopy groups v −1 1 π * (X ; p) of a topological space X and proved that if X is a sphere or compact Lie group, such as SU(n), each group v −1 1 π i (X ; p) is a direct summand of some actual homotopy group π j (X ). See also [7] for another expository account of v 1 -periodic homotopy theory.…”

Section: Outline Of Proofmentioning

confidence: 98%

A number-theoretic approach to homotopy exponents of SU(n)

Journal of Pure and Applied Algebra

Sun

2007

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“…Because of the mammoth nature of [2], we guide the reader to the relevant results. Referring always to [2], the specific statements regarding ν(sv There is a subtlety here for SO (9) and SO(10) which will be discussed at the beginning of Section 3. It is too technical to include in this introduction.…”

Section: Proposition 15 If ν(Svmentioning

confidence: 99%

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

Bendersky

Trans. Amer. Math. Soc.

Mahowald

2005

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Abstract. In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order 2 e over RP n if n is even and sufficiently large and e ≥ 75. In this paper, we use the Bendersky-Davis computation of v −1 1 π * (SO(m)) to show that the 1981 result extends to all e ≥ 5 (still provided that n is sufficiently large). If e ≤ 4, the result is often different due to anomalies in the formula for v −1 1 π * (SO(m)) when m ≤ 8, but we also determine the stable geometric dimension in these cases. Statement of resultsThe geometric dimension gd(θ) of a stable vector bundle θ over a space X is the smallest integer m such that θ is stably equivalent to an m-plane bundle. Equivalently, gd(θ) is the smallest m such that the classifying map X θ −→ BO factors through BO(m). The group KO(P n ) of equivalence classes of stable vector bundles over real projective space is a finite cyclic 2-group generated by the Hopf line bundle ξ n . Many papers (e.g., [1], [22], [23], [24]) have been devoted to computing the geometric dimension of multiples kξ n of the Hopf bundles, in part because certain cases are equivalent to determining whether P n can be immersed in a certain Euclidean space (e.g., [10]).In this paper, we prove the following theorem, which extends and completes a program initiated in [12]. It says that, for sufficiently large even n, the geometric dimension of a vector bundle over P n depends only on its order in KO(P n ) and the mod 8 value of n. Theorem 1.1. Let n = 2, 4, 6, or 8, and e ≥ 1.(1) There is an integer sgd(n, e) which equals the geometric dimension of all bundles of order 2 e in KO(P n ) for sufficiently large n satisfying n ≡ n mod 8. (2) If e ≥ 5, then sgd(n, e) = 2e + δ(n, e), where δ(n, e) is defined by Table 1.2.