Geometric Structures on the Cochains of a Manifold
Geometric Topology
2022-02-01 v5 Mathematical Physics
Algebraic Topology
math.MP
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.
Cite
@article{arxiv.math/0505227,
title = {Geometric Structures on the Cochains of a Manifold},
author = {Scott O. Wilson},
journal= {arXiv preprint arXiv:math/0505227},
year = {2022}
}
Comments
37 pages. Corrected typos. Labeled figures. Additional references and acknowledgements