Classifying Hilbert bundles

Abstract

A Hilbert bundle ( p, B, X) is a type of fibre space p: B → X such that each fibre p −1( x) is a Hilbert space. However, p −1( x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An ( m, n)-bundle over the pair ( X, A) is a Hilbert bundle ( p, B, X) such that the dimension of p −1( x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying ( m, n)-bundles over ( X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [ A, G m ( C n)]/[ X, U( n)] ¦ A, and the isomorphism classes of ( m, n)-bundles over ( X, A) which are trivial over X, A.