Completions and fibrations for topological monoids

Lima-Filho, Paulo

doi:10.1090/S0002-9947-1993-1134758-4

Completions and fibrations for topological monoids
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Trans. Amer. Math. Soc. 340 (1993), 127-147 Request permission

Abstract:

We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion ${M^ + }$ of a monoid $M$ in that class and the topologized Grothendieck group $\tilde M$ associated to $M$. 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 ${\text {CW}}$-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 $X \mapsto {\tilde {SP}}(X)$ in the category of finite ${\text {CW}}$-complexes (Dold-Thom theorem [DT56]) is obtained.

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