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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The stable geometric dimension of vector bundles over real projective spaces


Authors: Donald M. Davis, Sam Gitler and Mark Mahowald
Journal: Trans. Amer. Math. Soc. 268 (1981), 39-61
MSC: Primary 55N15; Secondary 55R25
MathSciNet review: 628445
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Abstract: An elementary argument shows that the geometric dimension of any vector bundle of order $ {2^e}$ over $ R{P^n}$ depends only on $ e$ and the residue of $ n\,\bmod \,8$ for $ n$ sufficiently large. In this paper we calculate this geometric dimension, which is approximately $ 2e$. The nonlifting results are easily obtained using the spectrum $ bJ$. The lifting results require $ bo$-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage $ bo$-resolutions act almost like $ K({Z_2})$-resolutions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0628445-8
PII: S 0002-9947(1981)0628445-8
Keywords: Geometric dimension of vector bundles, real projective space, obstruction theory, $ bo$-resolutions
Article copyright: © Copyright 1981 American Mathematical Society