Locally symmetric spaces and $K$–theory of number fields

Abstract

For a closed locally symmetric space $M=\Gamma \setminus G∕K$ and a representation $\rho :G\to GL\left(N,ℂ\right)$ we consider the pushforward of the fundamental class in $H_{\ast }\left(BGL\left(\overline{\mathbb{Q}}\right)\right)$ and a related invariant in $K_{\ast }\left(\overline{\mathbb{Q}}\right)\otimes \mathbb{Q}$. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of $ℝ$–rank one.