THE HOMOTOPY GROUPS OF A HOMOTOPY GROUP COMPLETION

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 left perpendiculer [S-k, M] right perpendiculer 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 pointwise 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.