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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 $$i<n/2$$.

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