Polyhedra for which every homotopy domination over itself is a homotopy equivalence

TL;DR

This paper investigates when homotopy domination implies homotopy equivalence in 2-dimensional polyhedra, proving it holds for broad classes of fundamental groups, thus advancing understanding of homotopy relations in geometric topology.

Contribution

It establishes that homotopy domination over itself implies homotopy equivalence for 2D polyhedra with certain fundamental groups, extending previous results and answering open questions.

Findings

No counterexamples among 2D polyhedra with soluble fundamental groups of cdG ≤ 2.

Homotopy types of quasi-homeomorphic ANRs are equal for these classes.

Results apply to hyperbolic and limit groups with finite aspherical presentations.

Abstract

We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for manifolds (Bernstein-Ganea, 1959), $1$-dimensional polyhedra and polyhedra with polycyclic-by-finite fundamental groups (DK, 2005). Thus one may ask, if there exists a counterexample among $2$-dimensional polyhedra with soluble fundamental groups. In this paper we show that it cannot be found in the class of $2$-dimensional polyhedra with soluble fundamental groups $G$ with cd$G \leq 2$ (and soluble can be replaced here by a wider class of elementary amenable groups). We prove more general fact, that there are no counterexamples among $2$-dimensional polyhedra, whose fundamental groups have finite aspherical presentations and are Hopfian (or more general, weakly Hopfian). In particular, a counterexample does not exist also among $2$-dimensional polyhedra whose fundamental groups are knot groups and in the class of $2$-dimensional polyhedra with one-related torsion-free Hopfian fundamental groups. The results can be applied also, for example, to hyperbolic groups or limit groups with finite aspherical presentations. For the same classes of polyhedra we get also a positive answer to another other open question: "Are the homotopy types of two quasi-homeomorphic ANR's equal?"