Papers › Persistent Mayer homology and persistent Mayer Laplacian
Persistent Mayer homology and persistent Mayer Laplacian
Li Shen, Jian Liu, Guo-Wei Wei
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In algebraic topology, the differential (i.e., boundary operator) typically satisfies d²=0. However, the generalized differential dᴺ=0 for an integer N≥2 has been studied in terms of Mayer homology on N-chain complexes for more than eighty years. We introduce Mayer Laplacians on N-chain complexes. We show that both Mayer homology and Mayer Laplacians offer considerable application potential, providing topological and geometric insights to spaces. We also introduce persistent Mayer homology and persistent Mayer Laplacians at various N. The Wasserstein distance and stability of persistence diagrams associated with Mayer homology are investigated. Our computational experiments indicate that the topological features offered by persistent Mayer homology and spectrum given by persistent Mayer Laplacians hold substantial promise for large, complex, and diverse data. We envision that the present work serves as an inaugural step towards integrating Mayer homology and Mayer Laplacians into the realm of topological data analysis.
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