Infinite dimensional manifolds from a new point of view

2010/12/29 by Xianzu, Lin
#18A05 #55U10 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1012.5885

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.

Related

• Chern-Weil theory for certain infinite-dimensional Lie groups
• Smooth simplicial sets and universal Chern-Weil for infinite dimensional groups
• Weil Spaces and Weil-Lie Groups
• Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups
• About extending the Chern-Weil classification to the simplicial principal bundles