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

Yasha Savelyev

TL;DR

The paper develops a robust framework of smooth simplicial sets to realize a universal differential graded Chern-Weil theory for infinite-dimensional generalized Lie groups. It constructs a smooth Kan model $BG^{\mathcal{U}}$ whose geometric realization matches Milnor's classifying space and defines a universal dg Chern-Weil homomorphism $cw: \mathcal{I}(G) \to \Omega^{\bullet}(BG^{\mathcal{U}}, \mathbb{R})$, with compatibility under pullbacks and classifying maps. The work culminates in a universal cohomological theory and a universal $dg$-Chern-Weil map, plus concrete applications to Hamiltonian group actions, Reznikov classes, and the coupling class for Hamiltonian fibrations; the constructions recover and extend classical CW theory to infinite dimensions and provide universal representatives for characteristic classes. This framework yields new, broadly applicable tools for studying characteristic classes of infinite-dimensional bundles and their geometric realizations.

Abstract

We give the construction of the universal, natural up to homotopy Chern-Weil differential graded algebra homomorphism: $$cw: \mathcal{I} (G) \to Ω^{\bullet } (BG, \mathbb{R})$$ for infinite dimensional Milnor regular Lie groups $G$, where $Ω^{\bullet}(BG, \mathbb{R})$ 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 $BG$.

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

TL;DR

The paper develops a robust framework of smooth simplicial sets to realize a universal differential graded Chern-Weil theory for infinite-dimensional generalized Lie groups. It constructs a smooth Kan model whose geometric realization matches Milnor's classifying space and defines a universal dg Chern-Weil homomorphism , with compatibility under pullbacks and classifying maps. The work culminates in a universal cohomological theory and a universal -Chern-Weil map, plus concrete applications to Hamiltonian group actions, Reznikov classes, and the coupling class for Hamiltonian fibrations; the constructions recover and extend classical CW theory to infinite dimensions and provide universal representatives for characteristic classes. This framework yields new, broadly applicable tools for studying characteristic classes of infinite-dimensional bundles and their geometric realizations.

Abstract

We give the construction of the universal, natural up to homotopy Chern-Weil differential graded algebra homomorphism: for infinite dimensional Milnor regular Lie groups , where is a certain de Rham algebra of (Milnor up to a natural weak homotopy equivalence) and where is the algebra of continuous, 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 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 as a smooth Kan complex, with the geometric realization weakly equivalent to the Milnor .
Paper Structure (44 sections, 37 theorems, 287 equations)

This paper contains 44 sections, 37 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. Generalized Lie groups
  3. Universal $dg$ Chern-Weil homomorphism
  4. Smooth structures on $BG$
  5. Cohomological Chern-Weil homomorphism
  6. $dg$ Enhancement
  7. An example: the group of Hamiltonian symplectomorphisms
  8. Other examples
  9. Acknowledgements
  10. Preliminaries and notation
  11. Topological simplices and smooth singular simplicial sets
  12. The simplex category of a simplicial set
  13. Geometric realization
  14. Smooth simplicial sets
  15. Smooth Kan complexes
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $G$ be a generalized Lie group then there is an algebra homomorphism: This has the following property. Suppose that $P \to Y$ is a smooth $G$-bundle over a smooth manifold $Y$, and is the associated Chern-Weil map. Then where is the classifying map of $P$ and the induced algebra map.

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7: Kedra-McDuff
  • Definition 2.1
  • Definition 2.3
  • ...and 93 more