The homotopy groups of a homotopy group completion
Let \(M\) be a topological monoid, \(BM\) the classifying space of \(M\), \(\Omega BM\) the (based) loop space of \(BM\), and \([S^k,M]\) the set of (unbased) homotopy classes of (continuous) maps between the \(k\)-sphere \(S^k\) and \(M\) (which is naturally a monoid). Say that \(M\) satisfies the \textit{strong homotopy commutativity hypothesis} (SHCH) if for every \(x\in M\) there exists a homotopy between the maps \((m,n)\mapsto mn\) and \((m,n)\mapsto nm\) that is constant on \((x,x)\). In this well-written and detailed paper, it is proved that if \(M\) satisfies SHCH, then for any \(k>0\), there is a natural isomorphism of groups: \[[S^k,M]/\pi_0(M)\cong \pi_k(\Omega BM).\] The isomorphism is described explicitly in the proof. A priori, it is not immediately clear to this reviewer that the quotient monoid \([S^k,M]/\pi_0(M)\) is a group to begin with; part of the proof of this result establishes the existence of inverses in \([S^k,M]/\pi_0(M)\). The result seems both powerful and interesting. Indeed, the author applies this theorem to numerous settings where topological monoids naturally arise and have been the subject of active study: the Lawson homology of a complex projective variety, the deformation representation ring of discrete groups, the deformation \(K\)-theory of discrete groups, and lastly to the lifting problem of (spherical) families of characters to families of representations. With regard to this last application, let \(\Gamma\) be a finitely generated discrete group and let \(G\) be a connected real reductive Lie group. The \(G\)-character variety of \(\Gamma\) is the conjugation quotient space \(\mathrm{X}(\Gamma, G):=\mathrm{Hom}(\Gamma, G)^*/G\) where \(\mathrm{Hom}(\Gamma, G)^*\subset \mathrm{Hom}(\Gamma, G)\) is the subspace of group homomorphisms whose \(G\)-conjugation orbits are closed. A spherical family is a continuous map \(\chi:S^k\to \mathrm{X}(\Gamma, G)\). Assume now that \(G_n\) is either \(\mathrm{GL}(n,\mathbb{C})\) or \(\mathrm{U}(n)\) so direct sums of characters makes sense with respect to the abelian monoid \(\coprod_{n=0}^\infty\mathrm{X}(\Gamma,G_n)\). The family \(\chi\) is \textit{stably nullhomotopic} if there exists \(\chi_0\) so \(\chi\oplus \chi_0\) is nullhomotopic, it is \textit{geometric} if there is a family \(\rho:S^k\to \mathrm{Hom}(\Gamma, G_n)\) that projects to \(\chi\) (up to homotopy), and it is \textit{stably geometric} if there exists \(\chi_0\) so \(\chi\oplus \chi_0\) is geometric. Assume that \(G_n=\mathrm{U}(n)\), then Ramras shows that if \(k\leq 2\), every spherical family for any \(\Gamma\) is stably geometric. On the other hand, he also shows for any \(k\) that spherical families for \(2\)-dimensional crystallographic groups and also for fundamental groups of products of compact aspherical surfaces (possibly with boundary) are stably geometric. These examples include finitely generated free groups and free abelian groups as special cases. Using results of the reviewer and \textit{C. Florentino} [Math. Ann. 345, No. 2, 453--489 (2009; Zbl 1200.14093); Topology Appl. 173, 32--58 (2014; Zbl 1300.14045)], Ramras shows this latter result also holds for free groups and free abelian groups when \(G_n=\mathrm{GL}(n,\mathbb{C})\).
\(K\)-theory, topological monoid, homotopy groups, character variety
scientific article