Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres
Witt Extension Theorem, hermitian form, homotopy groups, finite-dimensional vector space
The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R , C , or H . This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW ∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π 1 and π 2 of the induced real, complex, and quaternionic pseudospheres.
Farenick, D. R. and Ramsey, C. "Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres", J. of Pure and Applied Algebra 212 (2008) 2358-2367.
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