Tsukuba Journal of Mathematics
- Tsukuba J. Math.
- Volume 37, Number 2 (2013), 207-257.
Holonomy groups in a topological connection theory
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.
Tsukuba J. Math., Volume 37, Number 2 (2013), 207-257.
First available in Project Euclid: 17 January 2014
Permanent link to this document
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)
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