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Non-Parallelizability of Grassmann Manifolds

Published online by Cambridge University Press:  20 November 2018

S. Trew  and
P. Zvengrowski
S. Trew
Affiliation:
University of Calgary Calgary, Alberta, Canada
P. Zvengrowski
Affiliation:
University of Calgary Calgary, Alberta, Canada
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Abstract

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The fact that no real Grassmann manifolds Gk(Rn) are parallelizable (or even stably parallelizable) except for the obvious cases G1R2≅S1, G1(R4)≅G3(R4) ≅ RP3, and G1(R8)≅ G7(R8) ≅ RP7 was first noted by Hiller and Stong. Their work in turn depends on induction and the work of Oproiu, who examined detailed calculations of Stiefel-Whitney classes for k = 2, 3. In this note we give a short proof of this result, using elementary results from K-theory, that also covers the complex and quaternionic Grassmann manifolds.

Keywords

55N1557N2553C30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 1 , 01 March 1984 , pp. 127 - 128
Copyright
Copyright © Canadian Mathematical Society 1984

References

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