Compute the homology of Hom(Z^n, U(m))

Determine the integral homology groups H_*(Hom(Z^n, U(m)); Z) for integers n ≥ 1 and m ≥ 1.

Background

A central motivation for the stable splitting is to make progress on the (integral) homology of spaces of commuting n-tuples in U(m). The splitting after inverting m! simplifies structure but eliminates torsion, leaving the integral homology problem unresolved.

The authors explicitly note that the homology of Hom(Z^n, U(m)) is still unknown, highlighting this as a standing problem motivating their work.