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Session S27 - Categories and Topology

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

Morse theory for group presentations

Ximena Fernandez

Swansea University, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Andrews--Curtis conjecture (1965) is one the most relevant open problems in low-dimensional topology, with roots in Whitehead’s simple homotopy theory and combinatorial group theory. It is closely related to other important problems in algebraic topology such as the Whitehead asphericity conjecture, the Zeeman conjecture and the Poincaré conjecture (now a theorem).

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.

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