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 Ψ1(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 3polytopes are shown not to extend to higher dimensions. For ≥ 4, there exist simple dpclytopes with d+8 facets whose boundary complex cannot be expressed as a refinement of the boundary complex of v. dpolytope with d+7 facets. For ≥ 4, there exist simple dpolytopes 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 dpolytopes 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 dsimplex; for d≥6, nonsimple dpolytopes with d+3 facets having the same property are constructed. The main positive result is that the boundary complex of a dpolytope with d+2 facets, (d+3 facets if d = 4,5), may be ex¬pressed as a refinement of the boundary complex of the dsimplex with any two preassigned vertices principal. Several conjectures are made, among them the following genera¬lization of Balinsky's theorem on the dconnectedness of the graph of a dpolytope. If di+...+dk = ds di E ε N, then between any two vertices of adpolytepe exist strong chains of difaces, I = 1,...,k, disjoint except for the chosen vertices.
Type:  Thesis (Doctoral) 

Title:  Refinements in Boundary Complexes of Polytopes 
Open access status:  An open access version is available from UCL Discovery 
Language:  English 
Additional information:  Thesis digitised by EThOS. 
URI:  https://discovery.ucl.ac.uk/id/eprint/1572386 
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