Considerations about Andrews-Curtis invariants based on sliced 2-complexes

TL;DR

This paper explores algebraic invariants of sliced 2-complexes using topological relations, aiming to distinguish simple homotopy from 3-deformations and potentially challenge the Andrews-Curtis conjecture.

Contribution

It provides a complete geometric list of topological relations for sliced 2-complexes and introduces an additional relation crucial for algebraic invariants under 3-deformations.

Findings

Extended Quinn relations with a new relation for slicing changes

Demonstrated the new relation holds in algebraic TQFT calculations

Results applicable to 2-complexes with two generators and relators

Abstract

We consider a 2-complex in a particular form, called the Quinn model of a 2-complex. It can be sliced in graphs, where a change from one graph to another can be organized by a sequence of local transitions, which are described in a list of F. Quinn [Q1]. The decomposition of that 2-complex into graphs has to be translated into an algebraic context (for example Topological Quantum field theory (TQFT)) to construct suitable potential invariants under 3-deformations. These invariants are accessible for computation by using a supercomputer and the results may yield a counterexample to the Andrews-Curtis conjecture. To achieve invariance under 3-deformations, there are obvious topological relations among the local transitions, for example to deform a bubble out of a rectangle. In this paper our main result is that we contribute a complete list of such topological relations in a totally geometric fashion. One outcome of our considerations is that the corresponding list of F. Quinn [Q1] is extended by an additional relation which takes care of locally changing a slicing. We do not know so far whether this relation is a consequence of the remaining ones. But it may be crucial for further work to focus on such subtleties, as algebraic "simplifications", where this question is bypassed, so far have been unable to distinguish between simple homotopy and 3-deformations at all. In our introduction we summarize some known results on the situation when passing to Algebra; and in {\S} 8 we calculate an example of an algebraic TQFT in order to demonstrate that our additional relation holds. All considerations are carried out for 2-complexes with two generators and two defining relators. But the results also hold in the general case.