Hodge theory is a beautiful synthesis of geometry, topology, and analysis which has been developed in the setting of Riemannian manifolds. However, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step toward understanding the geometry of vision.Appendix B by Anthony Baker discusses a separable, compact metric space with infinite-dimensional α-scale homology.

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge–de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian.

It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains. Anal. Appl. 2015.13:291-301. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 04/09/15. For personal use only. N. Smalewith a corresponding triangle in a manifold of constant curvature. Such spaces were originally studied by Alexandrov [1], and developed further by Alexandrov and others [2]. Alexandrov spaces with curvature bounded above include compact Riemannian manifolds, but also Riemannian manifolds with boundary, as well as many singular spaces such as buildings, and certain simplicial complexes (references are given in Sec. 3). The main theorem proven here, is that if X is a compact Alexandrov space with curvature bounded above, with a Borel probability measure µ, and if α > 0 is sufficiently small, then the conditions given in [4] are satisfied and the corresponding Hodge theorem holds. In addition, the kernel of ∆ k,α is isomorphic to the real cohomology of X.In Sec. 2, we recall the basic framework and some facts on topology at a scale that were developed in [4] which are needed to state the main theorem, as well as to prove it. In Sec. 3, we summarize the basic definitions and properties of Alexandrov spaces with curvature bounded above. Section 4 is devoted to the statement and proof of the main theorem. Finally in Sec. 5, we compare the Hodge theorem given here as it applies to compact Riemannian manifolds with boundary to the classical Hodge theorem on these spaces, and we also give some simple examples of harmonic forms in this setting.The author would like to thank Thomas Schick for suggesting this problem, and the referee for numerous useful suggestions.

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