Generalizing the fundamental group (Q2762851)

Let \((M,a)\) be a connected manifold with base point \((a\in M)\). The paper proposes a generalization of the fundamental group of \(M\) and the universal covering bundle over \(M\). Starting with a pair \((E,C)\) consisting of a differential vector bundle \(E\) and a linear connection \(C\) in \(E\), the author constructs, via the holonomy representation of \(C\), a principal fiber bundle \(\Pi(M,a)=(Q'(a),p_a,M,G'(a))\). \(\Pi(M,a)\) is called the universal bundle of the pair \((M,a)\) and its structure group \(G'(a)\) is called the generalized fundamental group of \((M,a)\).NEWLINENEWLINENEWLINEEvery differential bundle \(\xi\) over \(M\) with structure group \(G\) is associated with \(\Pi(M,a)\) and with a continuous homorphism \(\rho: G'(a)\rightarrow G\). One obtains that a pair \((E,C)\) is determined, up to an isomorphism, by the holonomy representation of the connection \(C\) (Theorem 1).NEWLINENEWLINENEWLINEUsing some special subgroups of \(G'(a)\) the author gives a theorem for the construction of a differentiable pair \((E,C)\) by an interesting system of transition functions.