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Homological stability for classical groups over finite fields. (English) Zbl 0358.18013
Algebr. \(K\)-Theory, Proc. Conf. Evanston 1976, Lect. Notes Math. 551, 290-302 (1976).
[This article was published in the book announced in this Zbl 0328.00005.] Using etale homotopy theory and the well-known homological stability of the classifying spaces of the classical groups, one obtains a stability theorem of the form \(H_i(G_n(k), Z/q) \overset \sim \rightarrow H_i(G_{n+1}(k), Z/q)\) for \(G = GL, SL, U, Sp, 0, S0\); for \(k\) a finite field; for \(q\) a prime different from the characteristic of \(k\); and for \(i\leq g(n)\), where \(g(n)\) is a linear function depending only upon \(G\). This implies a stability theorem \(H_i(G_n(F), Z) \overset \sim \rightarrow H_i(G_{n+1}(F), Z)\) for \(F\) the algebraic closure of a finite field. Moreover, isomorphisms \(H_i(SL_n(k), Z/q) \overset \sim \rightarrow H_i(GL_{n}(K), Z/q)\) are proved for \(i<n/2\) and \(p\) the characteristic of a finite field \(k\). Together with Quillen’s stability theorem for \(H_i(GL_{n}(k), Z/q)\), these results imply isomorphisms \(H_i(SL_n(k), Z) \overset \sim \rightarrow H_i(SL_{n+1}(k), Z)\) for \(k\) a finite field with more than 2 elements and \(i<n/2\).

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
20J06 Cohomology of groups
15B33 Matrices over special rings (quaternions, finite fields, etc.)