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Remarks on the cohomological classification of certain Fréchet bundles. (English) Zbl 1063.58002

Summary: We discuss the classification of certain infinite-dimensional fiber bundles modelled on Fréchet spaces. First, we consider vector bundles (over a Banach manifold \(X\)) of fiber type a Fréchet space \(\mathbb{F}\), obtained as the limit of a projective system of Banach vector bundles. Such bundles are classified by the cohomology set \(H^1(X,\underline H^0(\mathbb{F}))\), with coefficients in the sheaf of germs of \(H^0(\mathbb{F})\)-valued smooth maps on \(X\), where \(H^0(\mathbb{F})\) is an appropriate topological Fréchet group replacing, the pathological \(\text{GL}(\mathbb{F})\). An analogous classification is proved for Fréchet principal bundles whose structural group can be realized as a projective limit of Banach Lie groups.

MSC:

58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58B25 Group structures and generalizations on infinite-dimensional manifolds
55R15 Classification of fiber spaces or bundles in algebraic topology
55N30 Sheaf cohomology in algebraic topology
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