Combinatorial and continuous Hodge theories
Józef Dodziuk · Bulletin of the American Mathematical Society · 1974
Let K be a finite simplicial complex.Eckmann (see [1]) observed that any inner product in cochain spaces C Q (K; R) gives rise to a combinatorial Hodge theory.The purpose of this note is to announce that if K is a smooth triangulation of a compact, oriented Riemannian manifold X, then the combinatorial Hodge theory (for a suitable choice of inner product in cochain spaces) is an approximation of the Hodge theory of forms on X.We wish to thank L. Bers, H. Garland, and I. M. Singer for their help in our research.* Research announcements, limited to 100 typed lines of 65 spaces each, may be submitted to those members of the Council whose names are marked by asterisks.Such announcements are intended to communicate outstanding results that are to be reported in full elsewhere.