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Some problems in combinatorial topology of flag complexes
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Adamaszek, Michał (2012) Some problems in combinatorial topology of flag complexes. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2585187~S1
Abstract
In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs.
In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space
of configurations of particles in the so-called hard-core models on various lattices.
We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general.
We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs.
The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Combinatorial topology, Combinatorial analysis, Graph theory, Flag manifolds | ||||
Official Date: | May 2012 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Jones, J. D. S. (John D. S.) | ||||
Sponsors: | University of Warwick. Centre for Discrete Mathematics and its Applications ; Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | vii, 158 leaves : illustrations | ||||
Language: | eng | ||||
Date of first compliant deposit: | 14 April 2020 | ||||
Date of first compliant Open Access: | 14 April 2020 |
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