Farenick, D.R.Ramsey, Christopher2020-10-082022-05-312022-05-312008Farenick, 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.https://hdl.handle.net/20.500.14078/1804The 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.enAll Rights ReservedWitt Extension Theoremhermitian formhomotopy groupsfinite-dimensional vector spaceArticlehttps://doi.org/10.1016/j.jpaa.2008.03.007