Tsukuba Journal of Mathematics

Holonomy groups in a topological connection theory

Kensaku Kitada

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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

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

First available in Project Euclid: 17 January 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

Slicing function direct connection parallel displacement holonomy group classification theorem


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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