Considerations for constructing Andrews-Curtis invariants of s-move 3-cells

TL;DR

This paper explores the construction of Andrews-Curtis invariants for s-move 3-cells by decomposing these cells into sequences of 2-cells, aiming to deepen understanding of simple homotopy equivalences.

Contribution

It introduces a method to decompose s-move 3-cells into 2-cell sequences, providing a framework for constructing Andrews-Curtis invariants in this context.

Findings

Decomposition of s-move 3-cells into 2-cell sequences

Sketches of proofs and ideas for invariant construction

Identification of problems for future research

Abstract

Two simple homotopy equivalent 2-complexes K2 and L2 are related by an algebraic criterion of their corresponding presentations as stated in [HoMeSier]. Frank Quinn set it into a topological context (see [Qu1]) and call these 2-complexes related by an s-move. Using elementary 3-expansions, K2 extends to 3-cells in K3 respectively L2 to 3-cells in L3. By associating both explorations we obtain a decomposition of the s-move 3-cells into a sequence of 2-cells. We present ideas, sketch of proofs and problems to use it for constructing an Andrews-Curtis invariant on s-move 3-cells.