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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The geometric dimension of some vector bundles over projective spaces

Authors: Donald M. Davis and Mark E. Mahowald
Journal: Trans. Amer. Math. Soc. 205 (1975), 295-315
MSC: Primary 55F25; Secondary 57D20
MathSciNet review: 0372854
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Abstract: We prove that in many cases the geometric dimension of the $ p$-fold Whitney sum $ p{H_k}$ of the Hopf bundle $ {H_k}$ over quaternionic projective space $ Q{P^k}$ is the smallest $ n$ such that for all $ i \leq k$ the reduction of the $ i$th symplectic Pontryagin class of $ p{H_k}$ to coefficients $ {\pi _{4i - 1}}(({\text{R}}{P^\infty }/{\text{R}}{P^{n - 1}})\Lambda bo)$ is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space $ {\text{R}}{P^{4k + 3}}$ in Euclidean space if the number of 1's in the binary expansion of $ k$ is between 5 and 8.

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Keywords: Geometric dimension, immersions of projective spaces, symplectic Pontryagin classes, connective $ K$-theory, modified Postnikov towers
Article copyright: © Copyright 1975 American Mathematical Society

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