Groupoids and computational topology

Abstract

This thesis contributes to the computational theory of finitely presented groupoids. It develops, implements and illustrates data types and algorithms aimed at pure and applied topology. In particular, the thesis designs and implements data types for:

• free groupoids, • elements in free groupoids, • finitely presented (fp) groupoids, • homomorphisms of fp groupoids.

The thesis designs and implements algorithms for:

• composition of elements in a free groupoid, • path components of a fp groupoid, • a finite presentation for the vertex group of a fp groupoid, • a finite presentation for finite index subgroups of an fp group, • pushouts of fp groupoids, • a finite presentation for the fundamental groupoid of a finite, regular CW- complex, • the homomorphism of fundamental fp groupoids induced by an inclusion of finite regular CW-complexes, • the low-dimensional cup product on the cohomology of a finite regular CW-complexes, • a re-implementation of the Mapper algorithm for obtaining examples of finite simplicial complexes derived from experimental data, • a re-implementation of an approximation for the dominant eigenvectors of a floating point symmetric matrix (for use with the Mapper algorithm).

The thesis contains illustrations of the above data types and algorithms such as:

• the computation of a finite presentation of the fundamental group of a finite regular CW-complex based on the groupoid version of the van-Kampen the- orem. This allows for parallel computation of low-dimensional cup products, • the fundamental groupoid (and group) of simplicial complexes arising, via Mapper, from gait analysis data, • the fundamental groupoid (and group) of simplicial complexes arising from time-series data.