#### Volume 12, issue 1 (2012)

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

### Thilo Kuessner

Algebraic & Geometric Topology 12 (2012) 155–213
##### 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 ${H}_{\ast }\left(BGL\left(\overline{ℚ}\right)\right)$ and a related invariant in ${K}_{\ast }\left(\overline{ℚ}\right)\otimes ℚ$. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric spaces of $ℝ$–rank one.

##### Keywords
symmetric spaces, algebraic $K$–theory, volume, Borel class
##### Mathematical Subject Classification 2000
Primary: 57R19, 53C35, 57M50
Secondary: 11R70, 22E46
##### Publication
Received: 15 November 2010
Revised: 18 November 2011
Accepted: 18 November 2011
Published: 24 February 2012
##### Authors
 Thilo Kuessner Korea Institute for Advanced Study, School of Mathematics Hoegi-ro 85 Dongdaemun-Gu Seoul 130-722 South Korea http://wwwmath.uni-muenster.de/u/kuessner/