Stability for infinite-dimensional fibre bundles

Abstract

In this paper, we prove that any locally trivial fibre bundles $p:X \to B$ with fibre $M$ a manifold modeled on an infinite-dimensional space $E$ (e.g. the Hilbert space $l_2$ or the Hilbert cube $Q$) is bundle isomorphic to the bundle $p * \mathrm{proj}:X \times E \to B$. Further, we can obtain a strong version of this Bundle Stability Theorem. From Bundle Stability Theorem, we can introduce the notion of deficiency in bundles. We show that a finite union of locally deficient sets is deficient and we prove a bundle version of Mapping Replacement Theorem.