A(infinity) Persistent Homology Estimates Detailed Topology from Pointcloud Datasets

Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set P subset of M of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of A(infinity)-persistent homology. In its most general case, this stability means that given a continuous function f :Y -> R on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of A(infinity)-barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes A(infinity)-persistence functorial.