The homology of special linear groups over polynomial rings

Abstract

We study the homology of SL n ( F[ t, t −1]) by examining the action of the group on a suitable simplicial complex. The E 1-term of the resulting spectral sequence is computed and the differential, d 1, is calculated in some special cases to yield information about the low-dimensional homology groups of SL n ( F[ t,t −1]). In particular, we show that if F is an infinite field, then H 2( SL n ( F[ t, t −1]), ℤ) = K 2( F[ t, t −1]) for n ≥ 3. We also prove an unstable analogue of homotopy invariance in algebraic K-theory; namely, if F is an infinite field, then the natural map SL n ( F) → SL n ( F[ t]) induces an isomorphism on integral homology for all n ≥ 2.