Renormalization groupoids in algebraic topology

Abstract

Continuing work begin in [19, 27], we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum M ξ in terms of characteristic numbers (indexed by quasisymmetric functions) for complex-oriented quasitoric manifolds, and show that automorphisms or cohomology operations on this representation are defined by the 'renormalization' Hopf algebra N * of formal diffeomorphisms at the origin of the noncommutative line, previously considered (over Q) in quantum electrodynamics [3].The resulting structure can be presented in purely algebraic terms, as a groupoid scheme over Z defined by a coaction of N * on the ring N * of noncommutative symmetric functions. We sketch some applications to symplectic toric manifolds, combinatorics of simplicial spheres, and statistical mechanics.