I will talk about some of the many applications of the Steinberg module in algebraic topology. The Steinberg module was first introduced in representation theory by Steinberg but later Solomon and Tits discovered a geometric description as the top homology of the Tits building. Borel and Serre found it to be the virtual dualizing module of arithmetic groups, which gives access to their high dimensional cohomology groups. It comes up in spectral sequences introduced by Quillen and by Rognes, respectively, that can be used to compute algebraic K-theory. If time permits, we will also see how the split Steinberg module can be used to improve the homological stability range of GL_n(Z) (concerning low dimensional homology of GL_n(Z)).

Some applications of the Steinberg module in algebraic topology Sponsored by the Meyer Fund