On $(K_\ast (\textbf {Z}/p^{2}\textbf {Z})$ and related homology groups

Abstract:

It is shown that, for $p \geqslant 5$, \[ R = {\mathbf {Z}} / {p^2}{\mathbf {Z}}, {K_3}(R) = {\mathbf {Z}} / {p^2}{\mathbf {Z}} + {\mathbf {Z}} / ({p^2} - 1){\mathbf {Z}}\] and ${K_4}(R) = 0$. Similar calculations are made for $R$ the ring of dual numbers over ${\mathbf {Z}} / p{\mathbf {Z}}$. The calculation reduces to finding homology groups of $\operatorname {Sl} (R)$. A key tool is the spectral sequence of the group extension of $\operatorname {Sl} (n, {p^2})$ over $\operatorname {Sl} (n, p)$. The terms of this spectral sequence depend in turn on the homology of $\operatorname {Gl} (n, p)$ with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of ${d^2}$.