On the Borsuk conjecture concerning homotopy domination

TL;DR

This paper investigates whether compact ANRs that mutually homotopy dominate are necessarily homotopy equivalent, providing conditions for positive answers and constructing counterexamples with specific properties.

Contribution

It offers new conditions based on fundamental groups for the Borsuk conjecture and constructs counterexamples of compact continua that are h-equal but not homotopy equivalent.

Findings

Conditions on fundamental groups can guarantee homotopy equivalence.

Existence of h-equal, non-homotopy equivalent continua with trivial algebraic invariants.

Construction of n-connected continua that are h-equal but not homotopy equivalent.

Abstract

In the seminal monograph "Theory of retracts", Borsuk raised the following question: suppose two compact ANR's are $h$--equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact $h$--equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology, homotopy groups etc.) The second class is given by $n$--connected continua, for any $n$, which are infinite $CW$--complexes, and hence ANR's, on a complement of a point.