Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908684742819840 |
|---|---|
| author | Savelyev, Yasha |
| author_facet | Savelyev, Yasha |
| contents | 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 $BG$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13272 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups Savelyev, Yasha Algebraic Topology Differential Geometry Symplectic Geometry 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 $BG$. |
| title | Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups |
| topic | Algebraic Topology Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2112.13272 |