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Products on spheres
Author(s) -
James I. M.
Publication year - 1959
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300001868
Subject(s) - mathematics , homotopy , spheres , bott periodicity theorem , unit (ring theory) , product (mathematics) , pure mathematics , homotopy group , manifold (fluid mechanics) , homotopy sphere , commutative property , unit sphere , algebra over a field , geometry , physics , astronomy , mechanical engineering , mathematics education , engineering
Summary 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.