Division Algebras and Topology
The preceding chapters examined the division algebras of the real numbers, the complex numbers, the quaternions and the octonions. These are of dimension 1, 2, 4 and 8, respectively. So far no algebraist has been able to show that every division algebra has to be of one of these four dimensions, though this surprising fact can be proved by topological methods. Hopf was able to prove in 1940 , that the dimension of a division algebra must be a power of 2. His proof, which used the homology groups of projective spaces, will be given in §1. In the year 1958, Kervaire and Milnor independently of one another proved that the power of 2 must be equal to 1, 2, 4 or 8 . They used for this purpose the periodicity theorem of Bott on the homotopy groups of unitary and orthogonal groups. The periodicity theorem had led to the development of K-theory (, ), a new cohomology theory with whose help many of the classical problems of topology, which had resisted the ordinary homology and cohomology theory, could be solved. We shall describe in §2 a proof of the (1, 2, 4, 8)-Theorem, which is based on K-theory.
KeywordsVector Bundle Projective Space Division Algebra Homology Class Cohomology Ring
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