Ximena Fernández 7/20/22: Morse theory for group presentations and the persistent fundamental group скачать в хорошем качестве

Ximena Fernández 7/20/22: Morse theory for group presentations and the persistent fundamental group

Discrete Morse theory is a combinatorial tool to simplify the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will present a refinement of this theory that guarantees not only a homotopy equivalence with the Morse CW-complex, but also a Whitehead simple homotopy equivalence. Moreover, it provides an explicit description of the attaching maps of the critical cells in the simplified complex and bounds on the dimension of the complexes involved in the deformation. This result provides the suitable theoretical framework for the study of different problems in combinatorial group theory and topological data analysis. I will show an application of this technique that allows to prove that some potential counterexamples to the Andrews-Curtis conjecture do satisfy the conjecture. Moreover, the method can also be extended to filtrations of CW-complexes, providing an efficient algorithm for the computation of the persistent fundamental group of point clouds in terms of group presentations. This is joint work with Kevin Piterman. Fernandez, X. Morse theory for group presentations. arXiv:1912.00115