2012/02/28 by Maia Fraser, Fraser, Maia
Computer Science · Mathematics · Medicine · #55U10 #62-07 #68R99 #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1202.6132
openalex publication_date 2012/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an extension to the simplicial Nerve Lemma which establishes isomorphism of persistent homology groups, in the case where the covering spaces are filtered. While persistent homology is now widely used in topological data analysis, the usual Nerve Lemma does not provide isomorphism of persistent homology groups. Our argument involves some homological algebra: the key point being that although the maps produced in the standard proof of the Nerve Lemma do not commute as maps of chain complexes, the maps they induce on homology do.