Some problems in combinatorial topology of flag complexes
Adamaszek, Michał (2012) Some problems in combinatorial topology of flag complexes. PhD thesis, University of Warwick.Full text not available from this repository.
Official URL: http://webcat.warwick.ac.uk/record=b2585187~S1
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
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
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Combinatorial topology|
|Official Date:||May 2012|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|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 (EPSRC) (EP/D063191/1)|
|Extent:||vii, 158 leaves : illustrations|
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