The L^p-index of the Hodge-Dirac operator on compact Riemannian manifolds

2025/12/27 by Cédric Arhancet, Arhancet, Cédric
Mathematics · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Geometry and complex manifolds #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2512.22517

openalex publication_date 2025/12/27 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28

Abstract

We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator D = d + d^* acting on the Banach space L^p(Ω^\bullet(M)) of differential forms over a compact Riemannian manifold M. Relying on the compactness of M, we establish that this operator is bisectorial and admits a bounded H^∞ functional calculus, without curvature assumptions. This result enables us to prove that the triple (C(M), L^p(Ω^\bullet(M)), D) constitutes a compact Banach spectral triple. We then investigate consistent pairings between the Banach K-homology and the K-theory of the algebra C(M), identifying the resulting Fredholm indices with classical topological invariants, and hence showing that they are independent of p. We recover the classical Euler characteristic and the Hirzebruch signature as L^p-indices, demonstrating the effectiveness of Banach noncommutative geometry for geometric analysis, beyond the Hilbertian setting.

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