Holonomy groups in a topological connection theory. (English) Zbl 1286.53030

Kitada, Kensaku

Tsukuba J. Math. 37, No. 2, 207-257 (2013).

In this paper the author defines, in the context of topological connection theory, slicing functions in topological bundles. In addition, the presented examples indicate that the slicing function is a generalization of the connection in the smooth category. Next, parallel displacements along admissible sequences, their holonomy groups and their fundamental properties are discussed. The author also defines a holonomy bundle of the parallel displacement and proves a holonomy reduction theorem and related results. In particular the category of principal bundles with parallel displacements over a fixed base space is studied. Assuming the existence of an initial object of a category of principal G-bundles, a classification theorem of topological principal G-bundles in terms of topological group homomorphisms is obtained. 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 as described by Milnor.

Reviewer: Corina Mohorianu (Iaşi)

MSC:

53C05	Connections (general theory)
53C29	Issues of holonomy in differential geometry
55R15	Classification of fiber spaces or bundles in algebraic topology
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