Change search
ReferencesLink to record
Permanent link

Direct link
Non-embeddability of geometric lattices and buildings
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2014. Vol. 51, no 4, 779-801 p.
Keyword [en]
Embedding, Buildings, Geometric lattice, Simplicial complex, Almost embedding, Finite projective space
National Category
Discrete Mathematics
URN: urn:nbn:se:kth:diva-140323DOI: 10.1007/s00454-014-9591-8ISI: 000337141000002ScopusID: 2-s2.0-84902269684OAI: diva2:689560
Knut and Alice Wallenberg Foundation, KAW 2005.0098

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

Available from: 2014-01-21 Created: 2014-01-21 Last updated: 2014-07-07Bibliographically approved
In thesis
1. Connectivity and embeddability of buildings and manifolds
Open this publication in new window or tab >>Connectivity and embeddability of buildings and manifolds
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.
TRITA-MAT-A, 2014:01
National Category
urn:nbn:se:kth:diva-140324 (URN)978-91-7501-992-5 (ISBN)
Public defence
2014-02-13, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Knut and Alice Wallenberg Foundation
Available from: 2014-01-22 Created: 2014-01-21 Last updated: 2014-01-22Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Vorwerk, Kathrin
By organisation
Mathematics (Div.)
In the same journal
Discrete & Computational Geometry
Discrete Mathematics

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 29 hits
ReferencesLink to record
Permanent link

Direct link