The homotopy groups of a homotopy group completion

TL;DR

This paper characterizes the homotopy groups of the homotopy group completion of a topological monoid under strong commutativity conditions, with applications to algebraic cycles and representation theory.

Contribution

It provides a concrete description of the homotopy groups of the monoid completion and applies this to Lawson homology and representation spaces.

Findings

Homotopy groups of the completion are quotients of free homotopy classes.

Application to Lawson homology links algebraic cycles with homotopy classes.

Results enable lifting of families of characters to representations.

Abstract

Let $M$ be a topological monoid with homotopy group completion $\Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $\pi_k (\Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.