dc.contributor.advisor Forman, Robin Trout, Aaron Dwight 2009-06-04T08:43:39Z 2009-06-04T08:43:39Z 2005 https://hdl.handle.net/1911/18824 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 [6] and Cheng [7]. 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, [10], 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. 47 p. application/pdf eng Mathematics Spaces with positive combinatorial curvature Thesis Text Mathematics Natural Sciences Rice University Doctoral Doctor of Philosophy Trout, Aaron Dwight. "Spaces with positive combinatorial curvature." (2005) Diss., Rice University. https://hdl.handle.net/1911/18824.
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