Marco Guerra

In applied topology, significant research efforts have been devoted to finding good representatives of topological features, especially focusing on geometric criteria of minimality in terms of volume or length. It has been noted in the past that, for manifold data, the Alexander isomorphism provides a link between codimension-1 cycles and connected components, for which a canonical notion of representative exists. However, in the simplicial case, Alexander duality holds only up to a barycentric subdivision, which makes it prohibitively expensive in practice. Consider a simplicial d-manifold obtained from a point sample; I will recall some established results on the computation of homology representatives, and then present a novel method to compute canonical (d-1)-dimensional cycle representatives via Alexander duality, by computing a smaller, relative barycentric subdivision, whose size is controllable. I will then move to some preliminary ideas and open questions concerning the problem of extending a simplicial manifold to a simplicial sphere. Joint work with Rémi Molinier.