On the homology and cohomology of congruence subgroups

Abstract

For any prime p let T’(p) denote the congruence subgroup of SL,@) of level p: T,(p) is the kernel of the surjective homomorphism SL,(iz)*SL,JFJ induced by the reduction modp ([FP is the field with p elements). The groups T,(p) are torsion-free for all odd primes p. Recent results of Charney [4] and Suslin [14] provide examples of coefficient groups M such that the groups T,(p) are homology stable with M-coefficients and that SL(IF,) acts trivially on HT M) : M= Q, z[l/p] or B/m (p not dividing m). For this choice of M we compare the stable homology groups of T,(p) with those of SL,(z) (Theorems 1.4 and 1.5). We then study the stable homology groups H;(T’(p); L/qd) (p#q primes, dr 1) for Osis5 (Section 2). Finally we look at the cohomology with 7?-coefficients and prove that the restriction homomorphism ZZ4(SL,(;2); Z)+H4(rn(p); Z) is zero for all odd primes p and n?9 (Corollary 3.3). This is an immediate consequence of the following result on the second Chern class of F”(p) : c,(T,,(p)) =0 for all odd primes p and n22.