scholarly journals 3-torsion in the Homology of Complexes of Graphs of Bounded Degree

2013 ◽  
Vol 65 (4) ◽  
pp. 843-862
Author(s):  
Jakob Jonsson
Keyword(s):  
Boundary Element ◽  
Simplicial Complex ◽  
Homology Class ◽  
Chain Complex ◽  
Semigroup Algebras ◽  
Bounded Degree ◽  
Matching Complex ◽  
Image Position

AbstractFor δ ≥ 1 and n ≥ 1, consider the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. This complex is of some importance in the theory of semigroup algebras. When δ = 1, we obtain the matching complex, for which it is known that there is 3-torsion in degree d of the homology whenever (n − 4)/3 ≤ d ≤ (n − 6)/2. This paper establishes similar bounds for δ ≥ 2. Specifically, there is 3-torsion in degree d wheneverThe procedure for detecting torsion is to construct an explicit cycle z that is easily seen to have the property that 3zis a boundary. Defining a homomorphism that sends z to a non-boundary element in the chain complex of a certain matching complex, we obtain that z itself is a non-boundary. In particular, the homology class of z has order 3.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

The volume of a lattice polyhedron

1963 ◽  
Vol 59 (4) ◽  
pp. 719-726 ◽  
Author(s):  
I. G. Macdonald
Keyword(s):  
Coordinate System ◽  
Simplicial Complex ◽  
Jordan Curve ◽  
Affine Plane ◽  
Unit Area ◽  
The Real ◽  
Real Affine ◽  
Image Position ◽  
Lattice Polyhedron

Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose 0-simplexes are points of L. Suppose X is pure and the frontier Ẋ of X is a Jordan curve; then there is a well-known formula for the area of X in terms of the number of points of L which lie in X and Ẋ respectively, namelywhere L(X) (resp. L(Ẋ)) is the number of points of L which lie in X (resp. Ẋ), and μ(X) is the area of X, normalized so that a fundamental parallelogram of L has unit area.

scholarly journals Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

2008 ◽  
Vol 17 (2) ◽  
pp. 225-238 ◽  
Author(s):  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Maximal Degree ◽  
Bounded Degree ◽  
Chromatic Polynomials ◽  
Image Position ◽  
Complex Zeros

We prove that the chromatic polynomial$P_\mathbb{G}(q)$of a finite graph$\mathbb{G}$of maximal degree Δ is free of zeros for |q| ≥C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

On a Theorem of Sullivan

1978 ◽  
Vol 21 (2) ◽  
pp. 201-206 ◽  
Author(s):  
Michael A. Penna
Keyword(s):  
Simplicial Complex ◽  
Barycentric Coordinates ◽  
Geometric Proof ◽  
Finite Simplicial Complex ◽  
Image Position

The purpose of this note is to give an elementary geometric proof of the following result stated by Sullivan (see (4)).Theorem 1 (Sullivan). Let K be a finite simplicial complex with vertices v1, …, vN and corresponding barycentric coordinates b1, …, bN. Then the algebra of rational PL forms on K

scholarly journals HIGHER STRING TOPOLOGY ON GENERAL SPACES

2006 ◽  
Vol 93 (2) ◽  
pp. 515-544 ◽  
Author(s):  
PO HU
Keyword(s):  
Simplicial Complex ◽  
Hochschild Cohomology ◽  
Chain Complex ◽  
Derived Category ◽  
Cochain Complex ◽  
String Topology ◽  
Sphere Bundle ◽  
Chain Complexes ◽  
Natural Notion ◽  
Finite Simplicial Complex

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

scholarly journals Largest Components in Random Hypergraphs

2018 ◽  
Vol 27 (5) ◽  
pp. 741-762 ◽  
Author(s):  
OLIVER COOLEY ◽  
MIHYUN KANG ◽  
YURY PERSON
Keyword(s):  
Connected Components ◽  
Bounded Degree ◽  
Uniform Hypergraphs ◽  
Original Contribution ◽  
Common Notion ◽  
Formula Group ◽  
The Common ◽  
Edge Probability ◽  
Image Position ◽  
Random Hypergraphs

In this paper we considerj-tuple-connected components in randomk-uniform hypergraphs (thej-tuple-connectedness relation can be defined by letting twoj-sets be connected if they lie in a common edge and considering the transitive closure; the casej= 1 corresponds to the common notion of vertex-connectedness). We show that the existence of aj-tuple-connected component containing Θ(nj)j-sets undergoes a phase transition and show that the threshold occurs at edge probability$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.Our main original contribution is abounded degree lemma, which controls the structure of the component grown in the search process.

scholarly journals Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

1995 ◽  
Vol 117 (2) ◽  
pp. 275-286 ◽  
Author(s):  
D. Kotschick ◽  
G. Matić
Keyword(s):  
Algebraic Curves ◽  
Homology Class ◽  
Algebraic Surfaces ◽  
Branched Covers ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces ◽  
Four Manifolds ◽  
Image Position ◽  
Simply Connected

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formulais the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

scholarly journals Matching Complexes of $3 \times n$ Grid Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Shuchita Goyal ◽  
Samir Shukla ◽  
Anurag Singh
Keyword(s):  
Simplicial Complex ◽  
Homotopy Equivalent ◽  
Grid Graphs ◽  
Comprehensive List ◽  
Matching Complex

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed  that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.  

scholarly journals Polylogarithmic bounds in the nilpotent Freiman theorem

2019 ◽  
pp. 1-17
Author(s):  
Matthew C. H. Tointon
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Linear Groups ◽  
Bounded Degree ◽  
Abelian Case ◽  
Optimal Bounds ◽  
Image Position

AbstractWe show that if A is a finite K-approximate subgroup of an s-step nilpotent group then there is a finite normal subgroup $H \subset {A^{{K^{{O_s}(1)}}}}$ modulo which ${A^{{O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)}}$ contains a nilprogression of rank at most ${O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K)$ and size at least $\exp ( - {O_s}(\mathop {\log }\nolimits^{{O_s}(1)} K))|A|$ . This partially generalises the close-to-optimal bounds obtained in the abelian case by Sanders, and improves the bounds and simplifies the exposition of an earlier result of the author. Combined with results of Breuillard–Green, Breuillard–Green–Tao, Gill–Helfgott–Pyber–Szabó, and the author, this leads to improved rank bounds in Freiman-type theorems in residually nilpotent groups and certain linear groups of bounded degree.

Injectivity of the Connecting Maps in AH Inductive Limit Systems

2005 ◽  
Vol 48 (1) ◽  
pp. 50-68 ◽  
Author(s):  
George A. Elliott ◽  
Guihua Gong ◽  
Liangqing Li
Keyword(s):  
Simplicial Complex ◽  
Inductive Limit ◽  
Extra Condition ◽  
The Third ◽  
Finite Simplicial Complex ◽  
Image Position ◽  
Metrizable Spaces ◽  
Special Case

AbstractLet A be the inductive limit of a systemwith , where Xn,i is a finite simplicial complex, and Pn,i is a projection in M[n,i](C(Xn,i)). In this paper, we will prove that A can be written as another inductive limitwith , where Yn,i is a finite simplicial complex, and Qn, i is a projection inM{n,i}(C(Yn,i)), with the extra condition that all the maps ψn,n+1 are injective. (The result is trivial if one allows the spaces Yn,i to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras. The special case that the spaces Xn,iare graphs is due to the third author.

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