2021/04/03 by Jacob Leygonie, Leygonie, Jacob, Ulrike Tillmann +1
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2104.01372
openalex publication_date 2021/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the inverse problem for persistent homology: For a fixed simplicial complex K, we analyse the fiber of the continuous map PH on the space of filters that assigns to a filter f: K → \mathbb R the total barcode of its associated sublevel set filtration of K. We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space, the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH-1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.