\(A^1\)-representability of hermitian \(K\)-theory and Witt groups
The author proves various results about hermitian \(K\)-theory and Balmer's Witt groups. One of the main results is the representability of hermitian \(K\)-theory and Witt groups both in the unstable and the stable \(A^1\)-homotopy category. This result is similar to the representability result for algebraic \(K\)-theory proved by \textit{V. Voevodsky} [``\(A^1\)-homotopy theory'', Doc. Math., J DMV, Extra Vol. ICM, Berlin 1998, Vol. I, 579--604 (1998; Zbl 0907.19002)] and \textit{F. Morel} and \textit{V. Voevodsky} [``\(A^1\)-homotopy theory of schemes'', Publ. Math. IHÉS 90, 45--143 (1999; Zbl 0983.14007)]. In fact, the author proves a general representability result based only on the Mayer-Vietoris property of Nisnevich squares and homotopy invariance. To establish these properties for the algebraically defined Balmer Witt groups the author shows that in fact these groups arise naturally as homotopy groups of a topological space. As applications of the results the author computes hermitian \(K\)-theory of the projective line and discusses the relationship to Morel's conjecture about \(A^1\)-homotopy groups of spheres.
scientific article; zbMATH DE number 2158645