Linear configurations of complete graphs K4 and K5 in ℝ3, and higher dimensional analogs

2017 ◽  
Vol 26 (05) ◽  
pp. 1750028
Author(s):  
Andrew Marshall
Keyword(s):  
Fundamental Group ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Simplicial Complexes ◽  
Alternating Group ◽  
Complete Graphs ◽  
Mapping Cylinder ◽  
Higher Dimensional ◽  
Special Cases

We investigate the space [Formula: see text] of images of linearly embedded finite simplicial complexes in [Formula: see text] isomorphic to a given complex [Formula: see text], focusing on two special cases: [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, and [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, so [Formula: see text] has codimension 2 in [Formula: see text], in both cases. The main result is that for [Formula: see text], [Formula: see text] (for either [Formula: see text]) deformation retracts to a subspace homeomorphic to the double mapping cylinder [Formula: see text] where [Formula: see text] is the alternating group and [Formula: see text] the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of the configuration space of points in the plane.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

A toral configuration space and regular semisimple conjugacy classes

1995 ◽  
Vol 118 (1) ◽  
pp. 105-113 ◽  
Author(s):  
G. I. Lehrer
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Cohomology Ring ◽  
General Setting ◽  
De Rham Cohomology ◽  
Pure Braid Group ◽  
De Rham ◽  
Image Position

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration spaceThe symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

scholarly journals Large random simplicial complexes, III the critical dimension

2017 ◽  
Vol 26 (02) ◽  
pp. 1740010 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Topological Properties ◽  
Open Questions ◽  
Random Simplicial Complexes ◽  
Special Cases

In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285 ; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837 ]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random [Formula: see text]-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

scholarly journals Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Prashanth Raman ◽  
Chi Zhang
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Moduli Space ◽  
Cluster Algebras ◽  
Configuration Spaces ◽  
Minkowski Sum ◽  
Canonical Forms ◽  
Infinite Class ◽  
Special Cases ◽  
Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

scholarly journals The Fundamental Group of Balanced Simplicial Complexes and Posets

10.37236/73 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Steven Klee
Keyword(s):  
Fundamental Group ◽  
Upper Bound ◽  
Large Family ◽  
Simplicial Complexes ◽  
Generating Set ◽  
Minimal Generating Set

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

scholarly journals Generalized Ramsey theory VI: Ramsey numbers for small plexes

1976 ◽  
Vol 22 (4) ◽  
pp. 400-410 ◽  
Author(s):  
Richard A. Duke ◽  
Frandk Harary
Keyword(s):  
Ramsey Theory ◽  
Simplicial Complexes ◽  
Ramsey Numbers ◽  
Higher Dimensional

AbstractGeneralized Ramsey theory for graphs was formulated and developed in the previous papers in this series. We extend the area here by introducing generalized Ramsey numbers for higher dimensional simplicial complexes. In particular we calculate explicitly the Ramsey numbers for several small “pure 2-complexes”, or more brieflyplexes, in which each edge is contained in some 2–call.

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

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