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A classification theorem for connections. (English) Zbl 0971.53019

The main goal in this paper is to generalize PoincarĂ©’s construction of universal coverings and fundamental groups in the realm of the geometric theory of connections. To this end, given a connected differentiable manifold \(M\) and a point \(a\) of \(M\), the author uses the theory of connections on bundles \((E,p,M)\) in order to define a certain group \(G'(a)\), called the generalized fundamental group of \((M,a)\) as well as a certain principal \(G'(a)\)-bundle \(\pi(M,a)\), called the universal bundle of \((M,a)\). The “universality” of this principal bundle is justified by proving that every differentiable finite dimensional bundle with structure group is associated to this universal bundle.

MSC:

53C05 Connections (general theory)
57M05 Fundamental group, presentations, free differential calculus
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