Geometric Structures on the Cochains of a Manifold

Details Geometric Structures on the Cochains of a Manifold Geometric Structures on the Cochains of a Manifold

2005-05-11

arxiv.org/abs/math/0505227v2

Mathematics

Geometric Topology

37 pages. Corrected typos. Labeled figures. Additional references and acknowledgements

Scientific paper

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 results on 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 limit as the mesh of the triangulation tends to zero.

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