Low dimensional spinor representations, Adams maps and geometric dimension

Lam, Kee Yuen; Randall, Duane

doi:10.1017/cbo9780511526305.008

Cited by 2 publications

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“…Since Sq 2 acts nontrivially on H 14 (RP 16 11 ; Z/2), we deduce that π 0 composed with the collapsing map RP 14 11 → S 14 is η: S 15 → S 14 . Consequently, π 0 represents a generator of π 15 (P 14 11 ) ≈ π 15 (V 15,4 ) = Z/8 by [20]. The order of (60α) + π 0 in π 15 (BSp(2)) is therefore a divisor of 8.…”

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 89%

“…The cellular structure of RP 14 11 is such that it can be realized as the mapping cone of a composite map S 12 2 e 13 → S 11 inclusion −−−−→ RP 12 11 where the first map extends η: S 12 → S 11 . Now 4 times the identity map of S 12 2 e 13 is trivial by [22].…”

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

“…Now 4 times the identity map of S 12 2 e 13 is trivial by [22]. Since 4 divides 60, the map (60α) + further extends to RP 14 11 → BSp(2), again denoted by (60α) + . We now take an extension RP 15 11 = RP 14 11 ∨ S 15 (60α )…”

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

“…The attaching map π: S 15 → RP 15 11 for the top-dimensional cell of RP 16 11 is the composition of the double covering S 15 → RP 15 with the collapsing map RP 15 c − → RP 15 11 . Homotopically, we can wedge decompose π as π 0 ∨ 2ι with π 0 : S 15 → RP 14 11 being the first component and the second component 2ι: S 15 → S 15 a map of degree 2. Since Sq 2 acts nontrivially on H 14 (RP 16 11 ; Z/2), we deduce that π 0 composed with the collapsing map RP 14 11 → S 14 is η: S 15 → S 14 .…”

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

“…Homotopically, we can wedge decompose π as π 0 ∨ 2ι with π 0 : S 15 → RP 14 11 being the first component and the second component 2ι: S 15 → S 15 a map of degree 2. Since Sq 2 acts nontrivially on H 14 (RP 16 11 ; Z/2), we deduce that π 0 composed with the collapsing map RP 14 11 → S 14 is η: S 15 → S 14 . Consequently, π 0 represents a generator of π 15 (P 14 11 ) ≈ π 15 (V 15,4 ) = Z/8 by [20].…”

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

See 4 more Smart Citations

Vector bundles of low geometric dimension over real projective spaces

Lam

Randall

2005

Math. Proc. Camb. Phil. Soc.

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Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 89%

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

Section: A 5-dimensional Spin Bundle ζ Over Rpmentioning

confidence: 99%

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Vector bundles of low geometric dimension over real projective spaces

Lam

Randall

2005

Math. Proc. Camb. Phil. Soc.

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

Bendersky

Davis

Mahowald

2005

Trans. Amer. Math. Soc.

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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.

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Low dimensional spinor representations, Adams maps and geometric dimension

Cited by 2 publications

References 0 publications

Vector bundles of low geometric dimension over real projective spaces

Vector bundles of low geometric dimension over real projective spaces

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

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