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.