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Title: Maps and hypermaps : operations and symmetry
Author: James, Lynne Denise
ISNI:       0000 0001 3588 836X
Awarding Body: University of Southampton
Current Institution: University of Southampton
Date of Award: 1985
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Just as a map on a surface is an imbedding of a topological realisation of a graph, a hypermap is an imbedding of a topological realisation of a hypergraph. The algebraic theory of (hyper)maps facilitates both a study of the symmetries of (hyper)maps and a study of the possible imbeddings of (hyper)graphs via an associated set of permutations. In chapter 1 we set out the established algebraic theory of maps on surfaces together with an extension to hypermaps and maps of higher dimension whose topological realisations include all cell decompositions of n-manifolds. There is a group of six invertible topological operations on surface maps which includes the well-known duality that interchanges vertices and faces. These operations arise naturally in the algebraic theory, being induced by the outer automorphisms of a certain Coxeter group. In chapter 2 we study the analogous groups of operations on hypermaps and maps of higher dimension. If the symmetry group of a map on a surface contains both a rotation centred on a face and a rotation centred on a vertex, each cyclically permuting successive incident edges, then the map is said to be regular. If, in addition, there is a symmetry which acts on an edge by interchanging the two incident vertices without interchanging the two incident faces then the map is said to be reflexible. In chapter 3 we consider a weaker version of these symmetry conditions, and in so doing we introduce a class of highly symmetric maps and hypermaps that remains invariant under the operations discussed in chapter 2. We find that every finitely generated group may be regarded as a group of symmetries of some highly symmetric hypermap. Finally, in chapter 4 we give an application of the algebraic theory to an imbedding problem by classifying those imbeddings of complete graphs whose symmetry group acts transitively on edges.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Algebraic theory of maps