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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
DOI: https://doi.org/10.1090/S0002-9947-1981-0628445-8
Correction: Trans. Amer. Math. Soc. 280 (1983), 841-843.
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 Mahowaldโ€™s theorem that beginning with the second stage $bo$-resolutions act almost like $K({Z_2})$-resolutions.


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Keywords: Geometric dimension of vector bundles, real projective space, obstruction theory, <IMG WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img22.gif" ALT="$bo$">-resolutions
Article copyright: © Copyright 1981 American Mathematical Society