Chern-Weil forms and abstract homotopy theory (Q2849009)

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.''