On the classification of real analytic fibre bundles (Q5947804)

The aim of this paper is to classify the real analytic \(G\)-bundles over an arbitrary analytic subspace \(X\) of \(\mathbb{R}^m\), \(G\) being a compact Lie group. So, first one introduces the class, \({\mathcal E}(X,G)\), of the analytic, principal, \(G\)-bundles over \(X\) which have the ``extension property'', that is which can be analytically extended to a smooth \(G\)-bundle over a manifold containing \(X\). Then, one shows that \({\mathcal E}(X,G)\) is the natural setting for the classification: in fact \({\mathcal E}(X,G)\) is exactly the class in which the topological equivalence always implies the analytical one and, more, there is a bijection between the family of the classes of analytically equivalent bundles of \({\mathcal E}(X,G)\) and \(H^1(X,{\mathcal G}_c)\), where \({\mathcal G}_c\) is the sheaf of germs of the continuous maps from \(X\) to \(G\). Moreover, if \(X\) is coherent one shows that \({\mathcal E}(X,G)\) is just the family of all analytic, principal, \(G\)-bundles over \(X\). The previous classification theorem has several consequences: some of these are here examined. Other classification theorems are obtained by using the family of the equivalence classes of analytically homotopic maps from \(X\) to a (suitable) classifying space. So, by means of these classes, one is able to give another classification for the bundles of \({\mathcal E}(X,G)\) and also to classify the bundles over a coherent space which are obtained from a given bundle by reducing its structure group \(G\) to a closed subgroup. Some of the preceding results lead to the notion of analytic universality: any (topologically) \(n\)-universal \(G\)-bundle \((G\) compact) appears to be \(n\)-universal with respect to the analytic bundles of \({\mathcal E}(X,G)\) and to the analytic maps between them. Last we remark that the key notion used is that of strongly analytic bundle; the complex theory is not employed.