A geometric Vietoris-Begle theorem, with an application to convex subsets of topological vector lattices

Abstract

We show that if L is a topological vector lattice, u:L→L is the function u(x)=x∨0, C⊂L is convex, and D=u(C) is metrizable, then D is an ANR and u|C:C→D is a homotopy equivalence, so D is contractible and thus an AR. This is proved by verifying the hypotheses of a second result: if X is a connected space that is homotopy equivalent to an ANR, Y is an ANR, and f:X→Y is a continuous surjection such that, for each y∈Y and each neighborhood V⊂Y of y, there is a neighborhood V′⊂V of y such that f−1(V′) can be contracted in f−1(V), then f is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.