Smooth singular complexes and diffeological principal bundles

Abstract

In previous papers, we used the standard simplices Δ ( ≥ 0) endowed with diffeologies having several good properties to introduce the singular complex D ( ) of a diffeological space . On the other hand, Hector and Christensen-Wu used the standard simplices Δ sub ( ≥ 0) endowed with the sub-diffeology of R +1 and the standard affine -spaces A ( ≥ 0) to introduce the singular complexes D sub ( ) and D aff ( ), respectively, of a diffeological space . In this paper, we prove that D ( ) is a fibrant approximation both of D sub ( ) and D aff ( ). This result easily implies that the homotopy groups of D sub ( ) and D aff ( ) are isomorphic to the smooth homotopy groups of , proving a conjecture of Christensen and Wu. Further, we characterize diffeological principal bundles (i.e., principal bundles in the sense of Iglesias-Zemmour) using the singular functor D aff . By using these results, we extend characteristic classes for D-numerable principal bundles to characteristic classes for diffeological principal bundles.