The Homology of Special Linear Groups over Polynomial Rings

We study the homology of SLn(F [t, t ]) by examining the action of the group on a suitable simplicial complex. The E–term of the resulting spectral sequence is computed and the differential, d, is calculated in some special cases to yield information about the low-dimensional homology groups of SLn(F [t, t ]). In particular, we show that if F is an infinite field, then H2(SLn(F [t, t ]), Z) = K2(F [t, t]) 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 SLn(F ) → SLn(F [t]) induces an isomorphism on integral homology for all n ≥ 2. Since Quillen’s definition of the higher algebraic K–groups of a ring [15], much attention has been focused upon studying the (co)homology of linear groups. There have been some successes—Quillen’s computation [14] of the mod l cohomology of GLn(Fq), Soulé’s results [18] on the cohomology of SL3(Z)—but few explicit calculations have been completed. Most known results concern the stabilization of the homology of linear groups. For example, van der Kallen [11], Charney [7], and others have proved quite general stability theorems for GLn of a ring. Also, Suslin [19] proved that if F is an infinite field, then the natural map Hi(GLm(F )) −→ Hi(GLn(F )) is an isomorphism for i ≤ m. Other noteworthy results include Borel’s computation of the stable cohomology of arithmetic groups [1], [2], the computation of H(SLn(F ), R) for F a number field by Borel and Yang [3], and Suslin’s isomorphism [20] of H3(SL2(F )) with the indecomposable part of K3(F ). This paper is concerned with studying the homology of linear groups defined over the polynomial rings F [t] and F [t, t]. One motivation for this is an attempt to find unstable analogues of the fundamental theorem of algebraic K–theory [15]: If R is a regular ring, then there are natural isomorphisms Ki(R[t]) ∼= Ki(R) (1) Supported by an Alfred P. Sloan Doctoral Dissertation Fellowship