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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript plus"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{M^ + }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a monoid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in that class and the topologized Grothendieck group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CW"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CW</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {CW}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the <italic>naïve</italic> approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow from bar ModifyingAbove upper S upper P With tilde left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mo stretchy="false"> ~ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X \mapsto {\tilde {SP}}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the category of finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="CW"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>CW</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {CW}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -complexes (Dold-Thom theorem [DT56]) is obtained.