Arithmetic Invariants of Subdivision of Complexes

1966 ◽  
Vol 18 ◽  
pp. 92-96 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Simplicial Complex ◽  
Euler Characteristic ◽  
Linear Combinations ◽  
Finite Simplicial Complex ◽  
Arithmetic Invariants

The following problem was raised by M. Brown. Let K be a finite simplicial complex, of dimension n, with αi(K) simplexes of dimension i. Which of the linear combinations have the property that they are unaltered by all stellar subdivisions of K? The most obvious invariant is the Euler characteristic; there are also some identities that hold for manifolds (2), so, if K is a manifold, they remain true on subdivision. We shall see that no other expressions are ever invariant, but if K resembles a manifold in codimensions ⩽2r (in a sense defined below) that r of the relations continue to hold.

Arithmetic Invariants of Simplicial Complexes

1980 ◽  
Vol 32 (6) ◽  
pp. 1306-1310
Author(s):  
M. Brown ◽  
A. G. Wasserman
Keyword(s):  
Simplicial Complex ◽  
Linear Function ◽  
Euler Characteristic ◽  
Simplicial Complexes ◽  
Homotopy Invariant ◽  
Finite Simplicial Complex ◽  
Arithmetic Invariants

What invariants of a finite simplicial complex K can be computed solely from the values v0(K), V1(K), …, vi(K), … where Vi(K) is the number of i-simplexes of K? The Euler chracteristic χ(K) = Σ i (– 1)ivi(K) is a subdivision invariant and a homotopy invariant while the dimension of K is a subdivision invariant and homeomorphism invariant. In [3], Wall has shown that the Euler chracteristic is the only linear function to the integers that is a subdivision invariant. In this paper we show that the only subdivision invariants (linear or not) of K are the Euler characteristic and the dimension. More precisely we prove the following theorem.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

Euler Classes of Combinatorial Manifolds

1980 ◽  
Vol 32 (4) ◽  
pp. 783-803
Author(s):  
Michael A. Penna
Keyword(s):  
Simplicial Complex ◽  
Large Class ◽  
Tangent Bundle ◽  
Finite Simplicial Complex

Every finite simplicial complex has a tangent bundle in the category of simplicial bundles (see [9]). The goal of this paper is to classify simplicial bundles, and, as an application of this result, to construct Euler classes for a large class of combinatorial manifolds. This construction is closely related to [3] and [4].

ℓ∞-COHOMOLOGY AND METABOLICITY OF NEGATIVELY CURVED COMPLEXES

1999 ◽  
Vol 09 (01) ◽  
pp. 51-77 ◽  
Author(s):  
IGOR MINEYEV
Keyword(s):  
Simplicial Complex ◽  
Universal Cover ◽  
Fundamental Groups ◽  
Sufficient Criterion ◽  
Positive Dimension ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Higher Dimensional ◽  
Isoperimetric Functions ◽  
Finite Simplicial Complex

We prove the analog of de Rham's theorem for ℓ∞-cohomology of the universal cover of a finite simplicial complex. A sufficient criterion is given for linearity of isoperimetric functions for filling cycles of any positive dimension over ℝ. This implies the linear higher dimensional isoperimetric inequalities for the fundamental groups of finite negatively curved complexes and of closed negatively curved manifolds. Also, these groups are ℝ-metabolic.

A Combinatorial Analogue of Poincaré's Duality Theorem

1964 ◽  
Vol 16 ◽  
pp. 517-531 ◽  
Author(s):  
Victor Klee
Keyword(s):  
Simplicial Complex ◽  
Upper Bound ◽  
Betti Number ◽  
Duality Theorem ◽  
Linear Inequalities ◽  
Negative Integer ◽  
Finite Simplicial Complex ◽  
Original Motivation

For a non-negative integer s and a finite simplicial complex K, let βS(K) denote the s-dimensional Betti number of K and let fs(K) denote the number of s-simplices of K. Our theorem, like Poincaré's, applies to combinatorial manifolds M, but it concerns the numbers fs(M) instead of the numbers βS(M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices of n-dimensional convex poly topes which have a given number i of (n — 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system of i linear inequalities in n variables, and was the original motivation for our study.

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 Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

2017 ◽  
Vol 60 (3) ◽  
pp. 470-477 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Simplicial Complex ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Homotopy Groups ◽  
Graded Lie Algebras ◽  
Simplicial Sets ◽  
Graded Lie Algebra ◽  
Finite Simplicial Complex

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

scholarly journals On Rham cohomology of locally trivial Lie groupoids over triangulated manifolds

2020 ◽  
Vol 13 (4) ◽  
pp. 116-125
Author(s):  
Jose R. Oliveira
Keyword(s):  
Simplicial Complex ◽  
Smooth Manifold ◽  
Piecewise Smooth ◽  
Ambient Space ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
De Rham Cohomology ◽  
Lie Groupoid ◽  
De Rham ◽  
Finite Simplicial Complex

Based on the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology of a transitive Lie algebroid, it is proved that the Rham cohomology of a locally trivial Lie groupoid G on a smooth manifold M is isomorphic to the piecewise Rham cohomology of G, in which G and M are manifolds without boundary and M is smoothly triangulated by a finite simplicial complex K such that, for each simplex ∆ of K, the inverse images of ∆ by the source and target mappings of G are transverses submanifolds in the ambient space G. As a consequence, it is shown that the piecewise de Rham cohomology of G does not depend on the triangulation of the base.

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