A NOTE ON THE HOMOTOPY TYPE OF BSL_3(Z)^2

By Ran Levi

To appear in the Mathematical Proceedings of the Cambridge Philosophical Society

Abstract

It is known that for p-perfect  groups G of finite virtual cohomological dimension 
and finite type mod-p cohomology, the p-completed classifying space BG^p has the 
property that \Omega BG^p is a retract of the loop space on a  simply-connected, Fp-
finite,   p-complete space. In this note we consider a particular example where this 
theorem applies, namely we study the homotopy type of  BSL_3(Z)^2. It  particular we 
analyze  \Omega BSt_3(Z)^2 a double cover of  \Omega BSL_3(Z)^2, and obtain a 
splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2^r 
maps on spheres. We also give a formula for  the Poincar\'e series of 
H_*(\Omega B\Gamma^p; Fp) for a general  group \Gamma, as above,  in terms of 
possibly simpler components. This formula is used to calculate the mod-2 homology of
\Omega B\Gamma^2 for \Gamma= SL_3(Z) or St_3(Z)^2  as modules over a certain 
tensor subalgebra.