A generalization of Steenrod’s approximation theorem

Christoph Wockel

Address: Fachbereich Mathematik, Technische Universität Darmstadt Schlossgartenstrasse 7, D-64289 Darmstadt, Germany

E-mail: christoph@wockel.eu

Abstract: In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous functions), preserving the section on regions where it is already smooth.

AMSclassification: primary 58B05; secondary 57R10, 57R12.

Keywords: infinite-dimensional manifold, infinite-dimensional smooth bundle, smoothing of continuous sections, density of smooth in continuous sections, topology on spaces of continuous functions.