0505227 (Scott O. Wilson)

Geometric Structures on the Cochains of a Manifold    [PDF]

Scott O. Wilson

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 holomorphic and anti-holomorphic cochains a combinatorial period matrix for surfaces. We show that for a particularly nice cochain inner product, several of these structures converge to their continuum analogues as the mesh of a triangulation tends to zero. It is an open question as to whether or not the combinatorial period matrix converges to the Riemann period matrix as the mesh of a sequence of triangulations tends to zero.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.

View original: http://arxiv.org/abs/0505227