V)$.

Topological Spaces of Cycles {#topspcyc}
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Group completions of monoids
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Let $M$ be a compactly generated Hausdorff topological abelian monoid. The *naive group completion* of $M$ is the quotient of $M\times M$ by the monoid action of $M$ where $M$ acts by $(a,b) \mapsto (m+a, m+b)$. Write $M^{+}$ for this abelian group, which is topologized as the quotient of $M\times M$. Recall that $M$ is said to satisfy the *cancellation property* if $a + m = b+m$ implies that $a = b$ for any $a,b,m\in M$. When $M$ satisfies cancellation then the naive group completion can be described as $M^{+}= M\times M/\sim$, where $(a,b) \sim (c,d)$ if $a+d = b +c$.

Naive group completion does not generally behave well topologically. For example it may happen that $M^{+}$ is not a Hausdorff topological group nor is it clear how homotopy invariants of $M$ and $M^{+}$ are related. Friedlander-Gabber [@FG:cyc] and Lima-Filho [@Li:completion] have studied conditions under which the naive group completion of a topological monoid is a Hausdorff group and $M\to M^{+}$ is a homotopy group completion. All of the topological monoids with which we work are tractable monoids in the sense of Friedlander-Gabber (see Appendix \[tract\]) and in 