On the generalized Andrews-Curtis-Problem -- A Disproof of the Relative Case

TL;DR

This paper investigates the generalized Andrews-Curtis Conjecture, demonstrating that certain complex deformations cannot be achieved when subcomplexes are fixed, thus providing a disproof of the relative case.

Contribution

It presents a disproof of the relative case of the generalized Andrews-Curtis Conjecture by showing the impossibility of certain 3-deformations with fixed subcomplexes.

Findings

Certain finite PLCW 2-complexes cannot be 3-deformed into each other when subcomplexes are fixed.

The conjecture fails in the relative case under specific conditions.

The result clarifies limitations in simple-homotopy equivalences with fixed subcomplexes.

Abstract

The generalized Andrews-Curtis Conjecture expects that finite PLCW 2-complexes which are simple-homotopy equivalent, can be 3-deformed into each other. If in addition subcomplexes are required to be kept fix during the deformation, this is not possible.