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Products on spheres

Published online by Cambridge University Press:  26 February 2010

I. M. James
The Mathematical Institute, 10 Parks Road, Oxford.
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Multiplications on spheres are studied in [11], [12] from the standpoint of homotopy theory. These multiplications are products with a unit element. The present paper deals with products in general. The investigation involves proving some results on the toric construction and the Whitehead product. These results also lead to theorems about the Stiefel manifold of unit tangent vectors to a sphere, originally proved by M. G. Barratt, which clear up some points in the homotopy theory of sphere bundles over spheres (see [15], [16]). They also enable us to prove that certain of the classical Lie groups are not homotopy-commutative.

Research Article
Copyright © University College London 1959

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