2022/12/06 by Oscar Harr, Harr, Oscar
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2212.03213
In this article, we extend an argument of Vogtmann in order to show homology stability of the Euclidean orthogonal group On(A) when A is a valuation ring subject to arithmetic conditions on either its residue or its quotient field. In particular, it is shown that if A is a henselian valuation ring, then the groups On(A) exhibit homology stability if the residue field of A has finite Pythagoras number. Our results include those of Vogtmann, and hold with various twisted coefficients. Using these results, we give analogues for fields F≠\mathbb R of some computations that appear in the study of scissor congruences.