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  • 1.
    Björner, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Connectivity of chamber graphs of buildings and related complexes2010In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 31, no 8, p. 2149-2160Article in journal (Refereed)
    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.

  • 2.
    Björner, Anders
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the connectivity of manifold graphs2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 10, p. 4123-4132Article in journal (Refereed)
    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.

  • 3.
    Hultman, Axel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Pattern avoidance and Boolean elements in the Bruhat order on involutions2009In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 30, no 1, p. 87-102Article in journal (Refereed)
    Abstract [en]

    We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.

  • 4. Tancer, Martin
    et al.
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-embeddability of geometric lattices and buildings2014In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444, Vol. 51, no 4, p. 779-801Article in journal (Refereed)
    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.

  • 5.
    Vorwerk, Kathrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Connectivity and embeddability of buildings and manifolds2014Doctoral 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.

    Download full text (pdf)
    Thesis
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