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Connectivity and embeddability of buildings and manifolds
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The results presented in is thesis concern combinatorial and topological properties of objects closely related to geometry, but regarded in combinatorial terms. Papers A and C have in common that they are intended to study properties of buildings, whereas Papers A and B both are concerned with the connectivity of graphs of simplicial complexes.

In Paper A it is shown that graphs of thick, locally finite and 2-spherical buildings have the highest possible connectivity given their regularity and maximal degree. Lower bounds on the connectivity are given also for graphs of order complexes of geometric lattices.

In Paper B an interpolation between two classical results on the connectivity of graphs of combinatorial manifolds is developed. The classical results are by Barnette for general combinatorial manifolds and by Athanasiadis for flag combinatorial manifolds. An invariant b Δof a combinatorial manifold Δ is introduced and it is shown thatthe graph of is (2dbΔ)-connected. The concept of banner triangulations of manifolds is defined. This is a generalization of flagtriangulations, preserving Athanasiadis’ connectivity bound.

In Paper C we study non-embeddability for order complexes of thick geometric lattices and some classes of finite buildings, all of which are d-dimensional order complexes of certain posets. They are shown to be hard to embed, which means that they cannot be embedded in Eucledian space of lower dimension than 2d+1, which is sufficient for all d-dimensional simplicial complexes. The notion of weakly independent atom configurations in general posets is introduced. Using properties of the van Kampen obstruction, it is shown that the existence of such a configuration makes the order complex of a poset hard to embed.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. , viii, 23 p.
Series
TRITA-MAT-A, 2014:01
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-140324ISBN: 978-91-7501-992-5 (print)OAI: oai:DiVA.org:kth-140324DiVA: diva2:689576
Public defence
2014-02-13, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Funder
Knut and Alice Wallenberg Foundation
Available from: 2014-01-22 Created: 2014-01-21 Last updated: 2014-01-22Bibliographically approved
List of papers
1. Connectivity of chamber graphs of buildings and related complexes
Open this publication in new window or tab >>Connectivity of chamber graphs of buildings and related complexes
2010 (English)In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 31, no 8, 2149-2160 p.Article in journal (Refereed) Published
Abstract [en]

Let Delta be a thick and locally finite building with the property that no edge of the associated Coxerer diagram has label "infinity". The chamber graph G(Delta), whose edges are the pairs of adjacent chambers in Delta is known to be q-regular for a certain number q = q(Delta). Our main result is that G(Delta) is q-connected in the sense of graph theory. In the language of building theory this means that every pair of chambers of Delta is connected by q pairwise disjoint galleries. Similar results are proved for the chamber graphs of Coxeter complexes and for order complexes of geometric lattices.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-26645 (URN)10.1016/j.ejc.2010.06.005 (DOI)000282674700017 ()2-s2.0-77956182341 (Scopus ID)
Note
QC 20101203Available from: 2010-12-03 Created: 2010-11-26 Last updated: 2017-12-12Bibliographically approved
2. On the connectivity of manifold graphs
Open this publication in new window or tab >>On the connectivity of manifold graphs
2015 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 10, 4123-4132 p.Article in journal (Refereed) Published
Abstract [en]

This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2015
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:kth:diva-140322 (URN)10.1090/proc/12415 (DOI)2-s2.0-84938252347 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20160602

Available from: 2014-01-21 Created: 2014-01-21 Last updated: 2017-12-06Bibliographically approved
3. Non-embeddability of geometric lattices and buildings
Open this publication in new window or tab >>Non-embeddability of geometric lattices and buildings
2014 (English)In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 51, no 4, 779-801 p.Article in journal (Refereed) Published
Abstract [en]

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such -dimensional complexes require -dimensional Euclidean space for an embedding. (This dimension is always sufficient for any -complex.) We develop a method to show non-embeddability for general order complexes of posets.

Keyword
Embedding, Buildings, Geometric lattice, Simplicial complex, Almost embedding, Finite projective space
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:kth:diva-140323 (URN)10.1007/s00454-014-9591-8 (DOI)000337141000002 ()2-s2.0-84902269684 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2005.0098
Note

Updated from "Submitted" to "Published". QC 20140707

Available from: 2014-01-21 Created: 2014-01-21 Last updated: 2017-12-06Bibliographically approved

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