## View abstract

### Session S27 - Categories and Topology

Friday, July 16, 13:30 ~ 14:00 UTC-3

## Morse theory for group presentations

### Ximena Fernandez

The Andrews--Curtis conjecture states that any balanced presentation of the trivial group can be transformed into the empty presentation through a sequence of a class of movements (called $Q^{**}$-transformations) that do not change its deficiency. The geometric equivalent formulation states that if $K$ is a contractible complex of dimension 2, then it 3-deforms to a point. Although the conjecture is known to be true for some classes of complexes, it still remains open.
In this talk, I will introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations. The procedure is based on a new version of discrete Morse theory that provides a simple homotopy equivalence between a given regular CW-complex and its Morse complex, with an explicit description of the attaching maps and bounds on the deformation. I will present applications of this technique to the study of potential counterexamples to the Andrews--Curtis conjecture, showing that many of them do satisfy the conjecture.