scholarly journals The homotopy theory of polyhedral products associated with flag complexes

2018 ◽  
Vol 155 (1) ◽  
pp. 206-228
Author(s):  
Taras Panov ◽  
Stephen Theriault
Keyword(s):  
Simplicial Complex ◽  
Homotopy Theory ◽  
Chordal Graph ◽  
Polyhedral Product ◽  
Flag Complex ◽  
Vertex Set ◽  
Flag Complexes

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co-$H$-space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

scholarly journals Simplicial Complexes are Game Complexes

10.37236/6958 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Sara Faridi ◽  
Svenja Huntemann ◽  
Richard J. Nowakowski
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Combinatorial Games ◽  
Natural Question ◽  
Flag Complex ◽  
Game Trees ◽  
Game Value ◽  
Flag Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

scholarly journals EFFICIENT DATA STRUCTURE FOR REPRESENTING AND SIMPLIFYING SIMPLICIAL COMPLEXES IN HIGH DIMENSIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 279-303 ◽  
Author(s):  
DOMINIQUE ATTALI ◽  
ANDRÉ LIEUTIER ◽  
DAVID SALINAS
Keyword(s):  
Data Structure ◽  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Nice Property ◽  
High Dimensions ◽  
Primary Interest ◽  
Flag Complex ◽  
Edge Contraction ◽  
Efficient Data ◽  
Flag Complexes

We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference Flag (G)\ K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.

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 $-1$.

Cup-products for the polyhedral product functor

2012 ◽  
Vol 153 (3) ◽  
pp. 457-469 ◽  
Author(s):  
A. BAHRI ◽  
M. BENDERSKY ◽  
F. R. COHEN ◽  
S. GITLER
Keyword(s):  
Simplicial Complex ◽  
Cohomology Ring ◽  
Decomposition Theorem ◽  
Ring Structure ◽  
Polyhedral Product ◽  
Cup Product ◽  
Rational Cohomology ◽  
Geometric Decomposition ◽  
Abstract Simplicial Complex ◽  
Cup Products

AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

scholarly journals Shellability and the Strong gcd-Condition

10.37236/67 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Alexander Berglund
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Flag Complex ◽  
Alexander Dual ◽  
Golod Ring

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.

scholarly journals The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem

2018 ◽  
Vol 2020 (11) ◽  
pp. 3364-3395 ◽  
Author(s):  
Alan Lew
Keyword(s):  
Simplicial Complex ◽  
General Position ◽  
Type Theorem ◽  
Spectral Gaps ◽  
Minimal Eigenvalue ◽  
Flag Complexes

Abstract Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{k}$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $\mu _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $\mu _{k}(X)$ for $k\geq d$ and $\mu _{d-1}(X)$. In particular, we establish the following vanishing result: if $\mu _{d-1}(X)>\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.

scholarly journals On Vietoris–Rips complexes of ellipses

2019 ◽  
Vol 11 (03) ◽  
pp. 661-690 ◽  
Author(s):  
Michał Adamaszek ◽  
Henry Adams ◽  
Samadwara Reddy
Keyword(s):  
Simplicial Complex ◽  
Finite Subset ◽  
Metric Space ◽  
Scale Parameter ◽  
Main Tool ◽  
Small Eccentricity ◽  
Scale Parameters ◽  
Homotopy Types ◽  
Rips Complex ◽  
Vertex Set

For [Formula: see text] a metric space and [Formula: see text] a scale parameter, the Vietoris–Rips simplicial complex [Formula: see text] (resp. [Formula: see text]) has [Formula: see text] as its vertex set, and a finite subset [Formula: see text] as a simplex whenever the diameter of [Formula: see text] is less than [Formula: see text] (resp. at most [Formula: see text]). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses [Formula: see text] of small eccentricity, meaning [Formula: see text]. Indeed, we show that there are constants [Formula: see text] such that for all [Formula: see text], we have [Formula: see text] and [Formula: see text], though only one of the two-spheres in [Formula: see text] is persistent. Furthermore, we show that for any scale parameter [Formula: see text], there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.

scholarly journals Spectral expansion of random sum complexes

2018 ◽  
Vol 12 (04) ◽  
pp. 989-1002
Author(s):  
Orr Beit-Aharon ◽  
Roy Meshulam
Keyword(s):  
Abelian Group ◽  
Simplicial Complex ◽  
Finite Abelian Group ◽  
Spectral Expansion ◽  
Random Sum ◽  
Random Subset ◽  
Minimal Eigenvalue ◽  
Vertex Set ◽  
Complex Valued ◽  
Dimensional Simplicial Complex

Let [Formula: see text] be a finite abelian group of order [Formula: see text] and let [Formula: see text] denote the [Formula: see text]-simplex on the vertex set [Formula: see text]. The sum complex [Formula: see text] associated to a subset [Formula: see text] and [Formula: see text], is the [Formula: see text]-dimensional simplicial complex obtained by taking the full [Formula: see text]-skeleton of [Formula: see text] together with all [Formula: see text]-subsets [Formula: see text] that satisfy [Formula: see text]. Let [Formula: see text] denote the space of complex-valued [Formula: see text]-cochains of [Formula: see text]. Let [Formula: see text] denote the reduced [Formula: see text]th Laplacian of [Formula: see text], and let [Formula: see text] be the minimal eigenvalue of [Formula: see text]. It is shown that if [Formula: see text] and [Formula: see text] are fixed, and [Formula: see text] is a random subset of [Formula: see text] of size [Formula: see text], then [Formula: see text]

scholarly journals Collapsibility of Non-Cover Complexes of Graphs

10.37236/8684 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ilkyoo Choi ◽  
Jinha Kim ◽  
Boram Park
Keyword(s):  
Simplicial Complex ◽  
Domination Number ◽  
Independent Set ◽  
Chordal Graphs ◽  
Alexander Dual ◽  
Vertex Set ◽  
Independence Complex ◽  
Independence Domination Number

Let $G$ be a graph on the vertex set $V$. A vertex subset $W \subseteq V$ is a cover of $G$ if $V \setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex on $V$ whose faces are non-covers of $G$. Then the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i\gamma(G)-1)$-collapsible where $i\gamma(G)$ denotes the independence domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs.

scholarly journals On the k-Buchsbaum property of powers of Stanley–Reisner ideals

2014 ◽  
Vol 213 ◽  
pp. 127-140 ◽  
Author(s):  
Nguyên Công Minh ◽  
Yukio Nakamura
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Residue Class ◽  
Simple Graph ◽  
Edge Ideal ◽  
Vertex Set

AbstractLet S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

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