Tsukuba Journal of Mathematics

Holonomy groups in a topological connection theory

Kensaku Kitada

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Abstract

We study slicing functions, which are called direct connections in the smooth category, and parallel displacements along sequences in a topological connection theory. We define holonomy groups for such parallel displacements, and prove a holonomy reduction theorem and related results. In particular, we study a category of principal bundles with parallel displacements over a fixed base space. Assuming the existence of an initial object of a category of principal $G$-bundles, we obtain a classification theorem of topological principal $G$-bundles in terms of topological group homomorphisms. It is shown that a certain object is an initial object if it is the holonomy reduction of itself with respect to the identification topology. The result is applied to the universal bundle over a countable simplicial complex constructed by Milnor.

Article information

Source
Tsukuba J. Math., Volume 37, Number 2 (2013), 207-257.

Dates
First available in Project Euclid: 17 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1389972028

Digital Object Identifier
doi:10.21099/tkbjm/1389972028

Mathematical Reviews number (MathSciNet)
MR3161576

Zentralblatt MATH identifier
1286.53030

Subjects
Primary: 53C05: Connections, general theory
Secondary: 53C29: Issues of holonomy 55R15: Classification 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

Keywords
Slicing function direct connection parallel displacement holonomy group classification theorem

Citation

Kitada, Kensaku. Holonomy groups in a topological connection theory. Tsukuba J. Math. 37 (2013), no. 2, 207--257. doi:10.21099/tkbjm/1389972028. https://projecteuclid.org/euclid.tkbjm/1389972028


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