Spaces with positive combinatorial curvature
Trout, Aaron Dwight
Doctor of Philosophy
We present two results concerning spaces with "positive combinatorial curvature". The first is analogous to the Bonnet-Myers theorem and the second to the maximum-diameter sphere theorems of Toponogov  and Cheng . We prove: (1) Any combinatorial 3-manifold whose edges have degree at most five has edge-diameter at most five. In higher dimensions, a combinatorial n-manifold whose (n - 2)-simplices have degree at most four has edge-diameter at most two. The fact that these degree bounds imply compactness was first proved via analytic arguments in a 1973 paper, , by David Stone. Our proof is completely combinatorial and provides sharp bounds for the edge-diameter of the triangulation. (2) Any M which satisfies the above hypotheses and has vertices v, w at the maximum edge-distance is a sphere. Moreover, the triangulation of M is entirely determined by Lk( v). That is, if M' is another n-manifold which satisfies our hypotheses and in which the v', w' have maximum edge-distance then any simplicial isomorphism Lk( v) ≅ Lk(v') extends to a simplicial isomorphism M ≅ M '. In fact, for each possible Lk(v ) which can appear we construct a sphere which satisfies our hypotheses and in which v and w have maximum edge-distance.