f Since S (X) = HomTo (t. £p_. e. a monotonic map of ordered sets, we first describe a n ~ n is a functor (~)op ---+Set, we will have shown that If f : m --+ n E IRm+l I sj~O, (t 0, ••• , tn) where t. 1 L f(j )=i sj, I: sj = 1} , let is a Y ---+ Hom 11 ~ ( ~ X) S(X) is a simplicial set. 2) ("The n-sphere"): is empty.

In particular, E E( R). We define a product on BGL(R), as follows. 4), the natural map k B(GL(R)xGL(R)/ is a homotopy equivalence. 8)) there is a map B (GL( R) x GL( R)) such that the induced map on fundamental groups carries E(R)x E(R) ~ BGL(R), into the 31 The Plus Construction E( R) C GL ( R), and hence induces a map E)+ between the plus commutator subgroup constructions . 9): (BGL(R)+, is a homotopy commutative and associative, +) H-space, hence a commutative connected H-~. 1he proof will depend on a few simple lemmas, which we prove first.

Then we have an automorphism fre~ hp: GLn(R) --+ GL(R) an inner conjugation on is the resulting map, then it is well defined upto GL(R). If iip: GLn(R) ~K 1 (R) composing with the natural quotient map, then depends only on the class ~xt, E(R)CGL(R), [ P 1 E K0 ( R). us one gets a well defined product defined upto an inner conjugation of E(R). -(hl'e rt* iip@ P = iip + iip • so and one checks that the induced map ·ib(R) (hp)• : H2 ( En( R), ZZ) - is induced by Hence hp ~ E(R) is well induces a map Further, one can check Hence the induced map depends only on the class [P1 t K0 ( R).