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Nonsingular bilinear maps revisited

Part of: Topological manifolds Forms and linear algebraic groups

Published online by Cambridge University Press:  17 March 2020

Carlos Domínguez  and
Kee Yuen Lam
Carlos Domínguez
Affiliation:
Académia de Matemáticas, Unidad Interdisciplinaria de Ingeniería Campus Guanajuato, Institúto Politécnico Nacional, Av. Mineral de Valenciana 200, Fracc. Industrial Puerto Interior, Silao de la Victoria, GuanajuatoC. P.36275, México (cdomingueza@ipn.mx)
Kee Yuen Lam
Affiliation:
Department of Mathematics, University of British Columbia, VancouverB.C., V6T 1Z2, Canada (lam@math.ubc.ca)
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Abstract

A bilinear map $\varPhi :\mathbb {R}^r\times \mathbb {R}^s\to \mathbb {R}^n$ is nonsingular if $\varPhi (\overrightarrow {a},\overrightarrow {b})=\overrightarrow {0}$ implies $\overrightarrow {a}=\overrightarrow {0}$ or $\overrightarrow {b}=\overrightarrow {0}$. These maps are of interest to topologists, and are instrumental for the study of vector bundles over real projective spaces. The main purpose of this paper is to produce examples of such maps in the range $24\leqslant r\leqslant 32,\ 24\leqslant s\leqslant 32,$ using the arithmetic of octonions (otherwise known as Cayley numbers) as an effective tool. While previous constructions in lower dimensional cases use ad hoc techniques, our construction follows a systematic procedure and subsumes those techniques into a uniform perspective.

Keywords

Bilinear mapsPolynomial productImmersionVector bundlesProjective spaces

MSC classification

Primary: 57N35: Embeddings and immersions
Secondary: 11E39: Bilinear and Hermitian forms
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 377 - 390
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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