  Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz
homomorphism for Baker and Richter's noncommutative complex cobordism spectrum
$M\xi$ in terms of characteristic numbers (indexed by quasi-symmetric
functions) for complex-oriented quasitoric manifolds, and show that
automorphisms or cohomology operations on this representation are defined by a
`renormalization' Hopf algebra of formal diffeomorphisms at the origin of the
noncommutative line, previously considered (over $Q$) in quantum
electrodynamics. The resulting structure can be presented in purely algebraic
terms, as a groupoid scheme over $Z$ defined by a coaction of this Hopf algebra
on the ring of noncommutative symmetric functions. We sketch some applications
to symplectic toric manifolds, combinatorics of simplicial spheres, and
statistical mechanics.
