On the 3-torsion in K4(Z)

Let SL4(Z) be the group of four by four integral matrices with determinant one. This group acts upon the top homology of the spherical Tits building of SL4 over Q, i.e. the Steinberg module St4 (see below, 1.2). The goal of this note is to prove the following: m 1. The first homology group H1(SL4(Z), St4) is a finite group of order a power of 2. esult was proved 18 years ago (Soulé, Thèse, University of Paris VII, 1979). At the time, I deduced from it that K4(Z) is the direct sum of a finite 2-group and 0 or Z/3. Rognes uses Theorem 1 in his proof that K4(Z) vanishes (J. Rognes, K4(Z) is the trivial group, Preprint, 1998).