Chern-Weil forms and abstract homotopy theory
type: Item
instance of: scholarly article
zbMATH Open document ID: 1281.58001
DOI: 10.1090/S0273-0979-2013-01415-0
author: Daniel S. Freed
author: Michael J. Hopkins
publication date: 2013-09-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.5959
Mathematics Subject Classification ID: 58A03
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C10
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 53C08
zbMATH DE Number: 6208218
zbMATH Keywords: Chern-Weil theory
zbMATH Keywords: homotopy
zbMATH Keywords: universal connection
zbMATH Keywords: universal space of differential forms
zbMATH Keywords: de Rham complex
MaRDI profile type: Publication
arXiv ID: 1301.5959
Wikidata QID: Q125945383
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title: Chern-Weil forms and abstract homotopy theory
published in: Bulletin of the American Mathematical Society. New Series
review text: This paper is devoted to the memory of Dan Quillen, whose works are well recognized in topology and geometry. As the authors mention in the last paragraph of the introduction, Dan Quillen introduced the abstract homotopy theory in [Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903); Ann. Math. (2) 90, 205--295 (1969; Zbl 0191.53702)] and also wrote about Chern-Weil theory in [Topology 24, 89--95 (1985; Zbl 0569.58030) and \textit{V. Mathai} and \textit{D. Quilllen}, Topology 25, 85--110 (1986; Zbl 0592.55015)].NEWLINENEWLINEThe underlying paper deals with Chern-Weil theory using differential homotopy theory. In the second section, the authors introduce the concept of universal connection \(U\nabla G\) on a certain principal bundle \(E_{\nabla} G\rightarrow B_{\nabla} G\) for a Lie group \(G\), the principal bundle being constructed here. For any triple of a differential manifold \(M\), a principal \(G\) bundle \(E\), a \(G\) connection \(\nabla\), there is a map \(f: M \rightarrow B_{\nabla} G\) such that \(\nabla\) is the pull back of \(U\nabla G\).NEWLINENEWLINEIn Problem 2.4 the authors post a question of computing the de Rham complex of \(B_{\nabla} G\), in the sense they defined, and state their result in Theorem 7.20, which is proven in this paper, especially in Section 8 with applying homotopy theory.NEWLINENEWLINEIn the abstract, the authors say: ``We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal \(G\)-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors.''
reviewed by: Daniel Guan
description: scientific article; zbMATH DE number 6208218