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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Completions and fibrations for topological monoids
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by Paulo Lima-Filho PDF
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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 127-147
  • MSC: Primary 55R35; Secondary 14C05, 55P10, 55R05, 55S15
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1134758-4
  • MathSciNet review: 1134758