Lockeberg, Erik Ring; (1978) Refinements in Boundary Complexes of Polytopes. Doctoral thesis , UCL (University College London).

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Abstract

A complex °K is said to be a refinement of a complex L if there exists a homeomorphism Ψ Set K set L such that for each face L of L Ψ-1(L) is a union of faces of K. A face K of K is said to be principal if Ψ(K) is a face of L. Some results concerning 3-polytopes are shown not to extend to higher dimensions. For ≥ 4, there exist simple d-pclytopes with d+8 facets whose boundary complex cannot be expressed as a refinement of the boundary complex of v. d-polytope with d+7 facets. For ≥ 4, there exist simple d-polytopes whose graphs do not con¬tain refinements of the complete graph on d+1 vertices, if three particular vertices are preassigned as principal. A conjecture of Grunbaum is answered in the negative by constructing, for ≥ 4 simple d-polytopes P with d+4 facets in which two particular vertices may not be preassigned as principal if the boundary complex of P is expressed as a refinement of the boundary complex of a d-simplex; for d≥6, non-simple d-polytopes with d+3 facets having the same property are constructed. The main positive result is that the boundary complex of a d-polytope with d+2 facets, (d+3 facets if d = 4,5), may be ex¬pressed as a refinement of the boundary complex of the d-simplex with any two preassigned vertices principal. Several conjectures are made, among them the following genera¬lization of Balinsky's theorem on the d-connectedness of the graph of a d-polytope. If di+...+dk = ds di E ε N, then between any two vertices of ad-polytepe exist strong chains of di-faces, I = 1,...,k, disjoint except for the chosen vertices.

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