Hereditary ball-covers for some Banach manifolds

Abstract:

At a problem seminar in Ithaca, New York, during January 1969, James Eells raised the question (numbered 33 on the circulated list) of whether a paracompact Fréchet manifold admits a locally finite cover by open sets, all of whose intersections are contractible. This had been established in the separable case by David Henderson, who obtained star-finite covers. This note settles the case that the model space is a Banach space homeomorphic to its countably infinite Cartesian power. The cover elements and all nonempty intersections are homeomorphic to the model. A short proof that the nerve of the cover has the homotopy type of the manifold is also included.