We illustrate an application of groupoids to topological data analysis. We consider a simplicial complex \(K\) with a map \(f\colon K \rightarrow M\) to a compact Riemannian manifold, equipped with its universal covering (e.g. a flat torus obtained by a quotient of the Euclidean plane). We discuss how a groupoid formalism allows us to computationally infer the induced homomorphism between the fundamental groups of \(K\) and \(M\) in terms of the group action of the universal covering. Since homology is the more computationally accessible invariant, we also show that we can obtain the induced homomorphism on \(H_1\) using a natural isomorphism between first singular homology and the first homology of the fundamental groupoid. 

We consider an application of this framework to point clouds in a compact Riemannian manifold, where we wish to deduce which cycles in a complex built on the point cloud correspond to "ambient" cycles in the manifold. This relies on associating a choice of minimising geodesics on \(M\) for each edge in the complex. We show that there is an open dense subset of the configuration space of the point cloud where this association can be uniquely made. We use this set up to empirically analyse and interpret the first principal persistence measures on different compact surfaces.