Infinite dimensional manifolds from a new point of view

Abstract

In this paper we propose a new treatment about infinite dimensional manifolds, using the language of category and functor. Our definition of infinite dimensional manifolds is a natural generalization of finite dimensional manifolds in the sense that de Rham cohomology and singular cohomology can be naturally defined and the basic properties (Functorial Property, Homotopy Invariant, Mayer-Vietoris Sequence) are preserved. In this setting we define the classifying space $BG$ of Lie group $G$ as an infinite dimensional manifold. Using simplicial homotopy theory and the Chern-Weil theory for principal $G$-bundles we show that de Rham's theorem holds for $BG$. Finally we get, as an unexpected byproduct, two new simplicial set models for the classifying spaces of compact Lie groups; it is totally different from the classical models constructed by Milnor Milgram, Segal and Steenrod.

arXiv commentthe definition of classifying spaces of Lie groups in the original paper ( Topology Appl. 159 (2012), no. 15, 3355--3362.)is corrected and all the results are preserved