Complex \(K\)-theory of \(B\text{SL}_ 3({\mathbb{Z}{}})\) (Q1207502)

The authors first study the Brown-Peterson (BP) cohomology of the classifying space of SL\(_ 3(\mathbb{Z})\); then, using the Conner-Floyd relation \[ K^*_{(p)}(-)\cong K^*\otimes_{\text{BP}^*}\text{BP}^*(-) \] they compute the complex \(K\)-theory of \(B\text{SL}_ 3(\mathbb{Z})\) and \(B\text{St}_ 3(\mathbb{Z})\), where St\(_ 3(\mathbb{Z})\) denotes the Steinberg group. Also, the authors prove that \(K^*(BG)\) is generated by Chern classes of complex representations, for \(G=\text{SL}_ 3(\mathbb{Z})\) and \(p=2\).