COMPLETIONS AND FIBRATIONS FOR TOPOLOGICAL MONOIDS

We show that, for a certain class of topological monoids, there is a h omotopy equivalence between the homotopy theoretic group completion M of a monoid M in that class and the topologized Grothendieck group M associated to M. The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic var iety and also the infinite symmetric products of finite CW-complexes. We associate principal fibrations to the completions of pairs of monoi ds, showing the existence of long exact sequences for the naive approa ch to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-S teenrod axioms for the functors X --> SP(X) in the category of finite CW-complexes (Dold-Thom theorem [DT56]) is obtained.