Approximate triangulations of Grassmann manifolds

TL;DR

This paper introduces a method for approximating the topology of Grassmann manifolds using nested simplicial complexes and persistent homology, enabling better understanding of their geometric structure.

Contribution

It defines approximate triangulations for manifolds and applies persistent homology to Grassmann manifolds, providing a new computational approach.

Findings

Successful construction of approximate triangulations for Grassmann manifolds

Demonstrated the use of persistent homology in identifying homological features

Provides a framework for topological analysis of embedded manifolds

Abstract

We define the notion of an approximate triangulation for a manifold $M$ embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in $M$ and use persistent homology to find a complex in the family whose homology agrees with that of $M$. Our key examples are various Grassmann manifolds $G_k({\mathbb R}^n)$.