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

 

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


Authors: Martin Bendersky, Donald M. Davis and Mark Mahowald
Journal: Trans. Amer. Math. Soc. 358 (2006), 1585-1603
MSC (2000): Primary 55S40, 55R50, 55T15
Published electronically: May 26, 2005
MathSciNet review: 2186987
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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 $e\ge75$. In this paper, we use the Bendersky-Davis computation of $v_1^{-1}\pi _*(SO(m))$ to show that the 1981 result extends to all $e\ge5$ (still provided that $n$ is sufficiently large). If $e\le4$, the result is often different due to anomalies in the formula for $v_1^{-1}\pi_*(SO(m))$ when $m\le8$, but we also determine the stable geometric dimension in these cases.


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Additional Information

Martin Bendersky
Affiliation: Department of Mathematics & Statistics, Hunter College, CUNY, New York, New York 10021
Email: mbenders@shiva.hunter.cuny.edu

Donald M. Davis
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
Email: dmd1@lehigh.edu

Mark Mahowald
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: mark@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03736-0
PII: S 0002-9947(05)03736-0
Keywords: Geometric dimension, vector bundles, homotopy groups
Received by editor(s): September 26, 2003
Received by editor(s) in revised form: May 20, 2004
Published electronically: May 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society