A Geometric Vietoris-Begle Theorem, with an Application to Riesz Spaces

Andrew McLennan

Published 2019-09-25Version 1

We show that a surjective map between compact ANR's (absolute neighborhood retracts) is a homotopy equivalence if the fibers are contractible and either the domain is simply connected or the fibers are also ANR's. This is a geometric analogue of the Vietoris-Begle theorem. We use it to show that if $L$ is a locally convex Riesz space, $C \subset L$ is compact, convex, and metrizable, $x \in L$, and the function $y \mapsto x \vee y$ ($y \mapsto x \wedge y$) is continuous, then the image of this map is contractible.