Smooth singular complexes and diffeological principal bundles

Kihara, Hiroshi

doi:10.2140/agt.2024.24.1913

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Smooth singular complexes and diffeological principal bundles

Hiroshi Kihara

Algebraic & Geometric Topology 24 (2024) 1913–1951

DOI: 10.2140/agt.2024.24.1913

Abstract

In previous papers, we used the standard simplices $Δ^{p}$ ( $p \geq 0$ ) endowed with diffeologies having several “good” properties to introduce the singular complex $S^{𝒟} (X)$ 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 $Δ_{sub}^{p}$ $(p \geq 0)$ endowed with the subdiffeology of $ℝ^{p + 1}$ and the standard affine $p$ –spaces $𝔸^{p}$ ( $p \geq 0$ ) to introduce the singular complexes $S_{sub}^{𝒟} (X)$ and $S_{aff}^{𝒟} (X)$ , respectively, of a diffeological space $X$ . We prove that $S^{𝒟} (X)$ is a fibrant approximation of both $S_{sub}^{𝒟} (X)$ and $S_{aff}^{𝒟} (X)$ . This result immediately implies that the homotopy groups of $S_{sub}^{𝒟} (X)$ and $S_{aff}^{𝒟} (X)$ 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 $S_{aff}^{𝒟}$ . By using these results, we extend the characteristic classes for $𝒟$ –numerable principal bundles to those for diffeological principal bundles.

Keywords

smooth singular complex, diffeological principal bundle

Mathematical Subject Classification

Primary: 58A40

Secondary: 18F15, 55U10

References

Publication

Received: 8 November 2021

Revised: 28 April 2023

Accepted: 11 May 2023

Published: 16 July 2024

Authors

Hiroshi Kihara
Center for Mathematical Sciences University of Aizu Aizuwakamatsu Japan

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Volume 24, issue 4 (2024)