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Completions and fibrations for topological monoids

Author: Paulo Lima-Filho
Journal: Trans. Amer. Math. Soc. 340 (1993), 127-147
MSC: Primary 55R35; Secondary 14C05, 55P10, 55R05, 55S15
MathSciNet review: 1134758
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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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Article copyright: © Copyright 1993 American Mathematical Society