Equivariant \(K\)-theory of central extensions and twisted equivariant \(K\)-theory: \(SL_{3}\mathbb{Z}\) and \(St_{3}{\mathbb{Z}}\) (Q303906)

Let \(G\) be a discrete group and let \(X\) be a proper \(G\)-CW complex. For any given \(\alpha \in H^3(X\times_GEG, \mathbb{Z})\) let \(^\alpha K^*_G(X)\) denote the twisted equivariant \(K\)-theory defined by the \textit{N. Bárcenas} et al. [Proc. Lond. Math. Soc. (3) 108, No. 5, 1313--1350 (2014; Zbl 1297.19008)]. Let \(\underline{E}G\) denote the universal proper \(G\)-CW complex. Then \(\underline{E}G\times_GEG\) gives a model for \(BG\) and hence the twisting for \(\underline{E}G\) is taken to be an element of \(H^3(BG, \mathbb{Z})\cong H^2(BG, S^1)\). In this paper the authors focus on the special linear group \(\mathrm{SL}_3\mathbb{Z}\) over integers. For its cohomology we know that \(H^3(\mathrm{SL}_3\mathbb{Z}, \mathbb{Z})\cong H^2(\mathrm{SL}_3\mathbb{Z}, S^1)\) is generated by two torsion elements \(u_1\), \(u_2\) and that \(u_1+u_2\) yields the central extension NEWLINE\[NEWLINE1 \to \mathbb{Z}/2 \to St_3\mathbb{Z} \to \mathrm{SL}_3\mathbb{Z} \to 1.NEWLINE\]NEWLINE The purpose of the paper is to compute all twisted equivariant \(K\)-groups of \(\mathrm{SL}_3\mathbb{Z}\). In fact, the authors obtain the following: NEWLINE\[NEWLINE\quad^{u_1}K^0_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) \cong\mathbb{Z}^{\oplus13}, ^{u_1}K^1_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) =0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE^{u_2}K^0_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) \cong\mathbb{Z}^{\oplus7},\quad^{u_2}K^0_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) =0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE^{u_1+u_2}K^0_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) \cong\mathbb{Z}^{\oplus 5}, ^{u_1+u_2}K^1_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}) \cong\mathbb{Z}/2. NEWLINE\]NEWLINE Here the first two formulas have been obtained previously by the authors themselves in [Algebr. Geom. Topol. 14, No. 2, 823--852 (2014; Zbl 1290.19002)]. As an application, in the final section (Section 7) the authors prove that these groups are isomorphic to the equivariant \(K\)-homology groups with coefficients in a \(\mathrm{SL}_3\mathbb{Z}\)-\(C^*\) algebra \(\mathcal{K}_\omega\) and also succeed, using last two formulas, in obtaining the completion of \(K^*_{St_3}(\underline{E}St_3\mathbb{Z})\) with respect to the augmentation ideal \(I_{St_3\mathbb{Z}}\). The proof of the remaining four formulas is done using a spectral sequence having as input the Bredon cohomology groups with coefficients in \(\alpha\)-twisted representations. For example, in the case of \(\alpha=u_2\) the authors obtain \(H^p_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}, \mathcal{R}_{u_2})=0\) for \(p>0\) and \(H^0_{\mathrm{SL}_3\mathbb{Z}}(\underline{E}\mathrm{SL}_3\mathbb{Z}, \mathcal{R}_{u_2}) \cong\mathbb{Z}^{\oplus 7}\) via a long calculation, which leads us to the collapsing of the spectral sequence and so to the desired result.