53 research outputs found
Tverberg's theorem with constraints
The topological Tverberg theorem claims that for any continuous map of the
(q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images
have a non-empty intersection. This has been proved for affine maps, and if
is a prime power, but not in general.
We extend the topological Tverberg theorem in the following way: Pairs of
vertices are forced to end up in different faces. This leads to the concept of
constraint graphs. In Tverberg's theorem with constraints, we come up with a
list of constraints graphs for the topological Tverberg theorem.
The proof is based on connectivity results of chessboard-type complexes.
Moreover, Tverberg's theorem with constraints implies new lower bounds for the
number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture
for , and .Comment: 16 pages, 12 figures. Accepted for publication in JCTA. Substantial
revision due to the referee
Shelling Coxeter-like Complexes and Sorting on Trees
In their work on `Coxeter-like complexes', Babson and Reiner introduced a
simplicial complex associated to each tree on nodes,
generalizing chessboard complexes and type A Coxeter complexes. They
conjectured that is -connected when the tree has
leaves. We provide a shelling for the -skeleton of , thereby
proving this conjecture.
In the process, we introduce notions of weak order and inversion functions on
the labellings of a tree which imply shellability of , and we
construct such inversion functions for a large enough class of trees to deduce
the aforementioned conjecture and also recover the shellability of chessboard
complexes with . We also prove that the existence or
nonexistence of an inversion function for a fixed tree governs which networks
with a tree structure admit greedy sorting algorithms by inversion elimination
and provide an inversion function for trees where each vertex has capacity at
least its degree minus one.Comment: 23 page
Five-Torsion in the Homology of the Matching Complex on 14 Vertices
J. L. Andersen proved that there is 5-torsion in the bottom nonvanishing
homology group of the simplicial complex of graphs of degree at most two on
seven vertices. We use this result to demonstrate that there is 5-torsion also
in the bottom nonvanishing homology group of the matching complex on
14 vertices. Combining our observation with results due to Bouc and to
Shareshian and Wachs, we conclude that the case is exceptional; for all
other , the torsion subgroup of the bottom nonvanishing homology group has
exponent three or is zero. The possibility remains that there is other torsion
than 3-torsion in higher-degree homology groups of when and .Comment: 11 page
The Brin-Thompson groups sV are of type F_\infty
We prove that the Brin-Thompson groups sV, also called higher dimensional
Thompson's groups, are of type F_\infty for all natural numbers s. This result
was previously shown for s up to 3, by considering the action of sV on a
naturally associated space. Our key step is to retract this space to a subspace
sX which is easier to analyze.Comment: Final version, in Pacific J. Math., 10 pages, 4 figure
- …