Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups

TL;DR

The paper constructs a universal Chern-Weil homomorphism for infinite-dimensional Lie groups, using smooth simplicial sets, and verifies a conjecture for Hamiltonian symplectomorphisms.

Contribution

It introduces a new geometric-categorical framework of smooth simplicial sets and constructs a universal Chern-Weil homomorphism for Milnor regular Lie groups.

Findings

Constructed a universal Chern-Weil homomorphism for infinite-dimensional groups.

Verified Reznikov's conjecture for Hamiltonian symplectomorphisms.

Developed a new approach to classifying spaces using smooth Kan complexes.

Abstract

We give the construction of the universal, natural up to homotopy Chern-Weil differential graded algebra homomorphism: $$cw: \mathcal{I} (G) \to \Omega ^{\bullet } (BG, \mathbb{R})$$ for infinite dimensional Milnor regular Lie groups $G$, where $\Omega ^{\bullet}(BG, \mathbb{R})$ is a certain de Rham algebra of $BG$ (Milnor $BG$ up to a natural weak homotopy equivalence) and where $\mathcal{I} (G)$ is the algebra of continuous, $Ad _{G}$ invariant, symmetric multilinear functionals on the Lie algebra. In particular, this applies to the group of compactly generated Hamiltonian symplectomorphisms, using which we verify a conjecture of Reznikov. For the construction of $cw$ we introduce a basic geometric-categorical notion of a smooth simplicial set. Loosely, this is to Chen spaces as simplicial sets are to spaces. We then give a new construction of the classifying space of $G$ as a smooth Kan complex, with the geometric realization weakly equivalent to the Milnor $BG$.