On the homology of special unitary groups over polynomial rings

Abstract

In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over ${\mathbb{P}}^1$, obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let $\mathcal{G}={\mathrm{SU}}_{3,{\mathbb{P}}^1}$ be the (non-isotrivial) non-split group-scheme over ${\mathbb{P}}^1$ defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism ${\mathrm{PGL}}_2(F)\to \mathcal{G}(F[t])$ that induces isomorphisms $H_{\ast }({\mathrm{PGL}}_2(F),\mathbb{Z})\to H_{\ast }(\mathcal{G}(F[t]),\mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$.

Comments are welcome