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 $H_*(\mathrm{PGL}_2(F), \mathbb{Z}) \to H_*(\mathcal{G}(F[t]), \mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$.