Smooth singular complexes and diffeological principal bundles

Kihara, Hiroshi¹

¹ Center for Mathematical Sciences, University of Aizu, Aizuwakamatsu, Japan

Algebraic and Geometric Topology, Tome 24 (2024) no. 4, pp. 1913-1951

Résumé

In previous papers, we used the standard simplices Δp (p ≥ 0) endowed with diffeologies having several “good” properties to introduce the singular complex S𝒟(X) of a diffeological space X. (Here, 𝒟 denotes the category of diffeological spaces.) On the other hand, Hector and Christensen–Wu used the standard simplices Δsub ⁡ p (p ≥ 0) endowed with the subdiffeology of ℝp+1 and the standard affine p–spaces 𝔸p (p ≥ 0) to introduce the singular complexes Ssub ⁡ 𝒟(X) and Saff ⁡ 𝒟(X), respectively, of a diffeological space X. We prove that S𝒟(X) is a fibrant approximation of both Ssub ⁡ 𝒟(X) and Saff ⁡ 𝒟(X). This result immediately implies that the homotopy groups of Ssub ⁡ 𝒟(X) and Saff ⁡ 𝒟(X) are isomorphic to the smooth homotopy groups of X, which enables us to give a positive answer to a conjecture of Christensen and Wu. Further, we characterize diffeological principal bundles (ie principal bundles in the sense of Iglesias-Zemmour) using the singular functor Saff ⁡ 𝒟. By using these results, we extend the characteristic classes for 𝒟–numerable principal bundles to those for diffeological principal bundles.

Détail

DOI : 10.2140/agt.2024.24.1913

Keywords: smooth singular complex, diffeological principal bundle