A Geometric Vietoris-Begle Theorem, with an Application to Convex Subsets of Topological Vector Lattices

TL;DR

This paper proves a geometric analogue of the Vietoris-Begle theorem, showing that certain convex subsets of topological vector lattices are homotopy equivalent to their images under a specific lattice operation, with applications to ANRs.

Contribution

It introduces a geometric Vietoris-Begle type theorem and applies it to convex subsets of topological vector lattices, establishing homotopy equivalences under specific conditions.

Findings

Convex subsets' images are ANRs.

The map u is a homotopy equivalence under certain conditions.

The theorem extends classical results to topological vector lattices.

Abstract

We show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence 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 $f \colon X \to Y$ is a continuous surjection such that for each $y \in Y$ and each neighborhood $V \subset Y$ of $y$, there is a neighborhood $V' \subset V$ 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.