Combinatorial Gauss-Bonnet theorem and the Alexander-Spanier cohomology
Abstract：For a smooth surface the celebrated Gauss-Bonnet theorem tells that the integration of the Gauss curvature is equal to the Euler number of surface times 2pi. Also on a combinatorial surface, namely a polyhedral surface, there is a similar theorem to the above, that is, the sum of Angle defect at each vertex amounts to the Euler number of surface times 2pi. This theorem goes back at least as far as Descartes. Thus the Angle defect seems to be a counterpart of the Gauss curvature on a polyhedral surface. In this talk we justify it by introducing a notion of the Alexander-Spanier cohomology, which also make possible a generalization of the theorem in higher dimensional case. The main subjects in my talk are polyhedrons, which are definitely accessible for everyone.