  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.
