Persistent intersection homology
Persistent homology uses a filtration on a topological space to define birth and death of a homology class. The \(r\)-dimensional classes that are born at stage \(i\) and die at stage \(j\) form the pair group \(P^{i,j}_r\), a vector space over the field with 2 elements, of the filtered space. There may be classes that never die, so-called essential classes. To force every class to die eventually, one uses a second filtration and relative homology to extend persistence and pair groups. If the space is an oriented manifold, and the filtrations are the sub- and superlevel sets of a Morse height function, then Poincaré duality induces a perfect pairing between the complementary pair groups \(P^{i,j}_r\) and \(P^{2n-j+1, 2n-i+1}_{d-r}\), where \(d\) is the dimension of the manifold and \(n\) is the number of critical points. Simple examples show that ordinary persistent homology is inadequate to detect important topological features when height functions on singular stratified spaces are studied. Also, the above pair group duality fails on such spaces. The paper under review defines persistent intersection homology to rectify these issues. The main result is as follows: Suppose that a stratified Morse function with \(n\) critical points is given on an oriented \(d\)-dimensional stratified space. Let \(\bar{p}\) and \(\bar{q}\) be complementary perversities. Then Poincaré-Lefschetz-Goresky-MacPherson duality for intersection homology induces a perfect pairing between the complementary intersection pair groups \(I^{\bar{p}} P^{i,j}_r\) and \(I^{\bar{q}} P^{2n-j+1, 2n-i+1}_{d-r}\) of the space filtered by the sub- and superlevel sets of the Morse function. Persistence diagrams and their symmetries are discussed as well. In fact, intersection homology persistence arises by specializing a more general concept of \(\phi\)-persistence, introduced in the present paper, where \(\phi\) is any binary function on a simplicial complex, defining the allowability of simplices. For filtered simplicial complexes, algorithms that compute \(\phi\)-persistence, in particular persistent intersection homology groups, are provided.
scientific article; zbMATH DE number 5908360
scientific article