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Hence BGL( R)+ is a 2-sided identity, upto homotopy, for the text, there exist x(t) y = w•• (ylt) x), and w, w' £ M such that for any x, y, z £ GL ( R), (x@ y)® z = w;. (x® (y® z)). homotopy commutative and associative. Since it is Hence connecte~ BGL(Rt is it is a H-group (lemma (A. 47 )) • LODAY'S PRODUCT IN K-1HEORY Following [L], we construct natural products Ki(R)@Kj(R) which generalize the product for ~ Ki+j(R), i,j~l i =j =1 First, fix a choice of a bijection R, R' and elements into an element of considered in §1 •

We set co, f(y)) - (F(f)o, y) for all Let yEL;m, where (o, f(y))E F(_n)x "'n·' and (F(f)o,y)cF(m) x "'nr be the equivalence relation so generated, and let IFI be the quotient space, with the quotient topology (an open subset is a set whose inverse image in l1 n F~) x 6 n is open). e. e. , Set) we get a continuous map any simplicial set F, a simplex O£ F (~) IF I ~ IGI For is called non-degenerate if it is not the degenerate simplex assigned to any (n-1)-simplex by one of the dege- Classifying Space 38 neracies.

Then a = (A~Im' A-l~I m' give commuting elements of {A, B} = E 3mn (R). S for details). 14) (Matsumoto): If F * {a, -a} = 1. a E R, is a (commutative) field, then K2 (F) has a presentation as the free abelian group on the symbols {a, b) with a, bE F * subject to the relations: "Classical" K-Theory 17 a} - 1 {a, b} = {b, (ii) for all {a, 1-a} (iii) for all a t . D Let F be a field. A symbol on F with values in an abelian group G ) F * x F * --+G is a map ( . (ii) (a, b) = ~latsumoto's values in (b, a) -1 for all (Tame Symbol) whose residue field is : U 1jJ k(v) * - .

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