Contractible polyhedra in products of trees and absolute retracts in products of dendrites

TL;DR

This paper characterizes when compact polyhedra can embed in products of trees and explores conditions under which absolute retracts can be embedded in products of dendrites, revealing both positive and negative embedding results.

Contribution

It provides a precise criterion for embedding polyhedra into products of trees and identifies limitations for embedding certain absolute retracts into products of dendrites.

Findings

A compact n-polyhedron embeds in a product of n trees iff it collapses onto an (n-1)-polyhedron.

Contractible n-polyhedra (n ≠ 3) can be embedded with the product of trees collapsing onto the image.

Existence of a 2-dimensional absolute retract that cannot embed in any product of 2+k dendrites when multiplied by I^k.

Abstract

We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n-1)-polyhedron. If the n-polyhedron is contractible and n\ne 3 (or n=3 and the Andrews-Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding. In contrast, there exists a 2-dimensional compact absolute retract X such that X\times I^k does not embed in any product of 2+k dendrites for each k.