Geometric Structures on the Cochains of a Manifold

Abstract

In this paper we develop several algebraic structures on the simplicial cochains of a triangulated manifold that are analogues of objects in differential geometry. We study a cochain product and prove several statements about its convergence to the wedge product on differential forms. Also, for cochains with an inner product, we define a combinatorial Hodge star operator, and describe some applications, including a combinatorial period matrix for surfaces. We show that for a particularly nice cochain inner product, these combinatorial structures converge to their continuum analogues as the mesh of a triangulation tends to zero.

arXiv comment37 pages. Corrected typos. Labeled figures. Additional references and acknowledgements

References in corpus (5)

• A geometric discretisation scheme applied to the Abelian Chern-Simons theory
• R-torsion and linking numbers from simplicial abelian gauge theories
• Discrete period matrices and related topics
• Finite size corrections for the Ising model on higher genus triangular lattices
• Discrete Polynomials and Discrete Holomorphic Approximation

Cited by in corpus (6)

• Transferring homotopy commutative algebraic structures
• A Gravitational Effective Action on a Finite Triangulation
• A gauge-invariant discrete analog of the Yang-Mills equations on a double complex
• Differential complexes and exterior calculus
• A Gravitational Effective Action on a Finite Triangulation as a Discrete Model of Continuous Concepts
• Discrete Connection Laplacians