Completions and fibrations for topological monoids

Abstract

We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion M + {M^ + } of a monoid M M in that class and the topologized Grothendieck group M ~ \tilde M associated to M M . The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite CW {\text {CW}} -complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naïve approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors X ↦ S P ~ ( X ) X \mapsto {\tilde {SP}}(X) in the category of finite CW {\text {CW}} -complexes (Dold-Thom theorem [DT56]) is obtained.