Questions tagged [simplicial-complexes]

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Self-dual simplicial complexes

A simplicial complex $K$ on a vertex set $[m] = \{1,...,m \}$ - where here we say vertex set to mean an ambient set of vertices on which $K$ is defined - is self-dual if it is equal to its Alexander ...
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1answer
140 views

Irreducible representations of the symmetric group on homology of simplicial complex

I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects: Consider an action of $\Sigma_k$ on a finite ...
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Enumeration and encoding of simplicial complexes

I'd like to know how to enumerate and encode all (abstract) simplicial complexes of a given kind. To start as simple as possible, consider the familiy $\mathcal{S}_n^{d}$ of simplicial complexes which ...
5
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1answer
186 views

Are there simplicial spheres with “non-geometric symmetries”?

Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$. ...
7
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1answer
186 views

What is a subdivision of an abstract simplicial complex?

I am looking for the definition of the subdivision of a simplicial complex. When the complex is defined in a geometric way, then the definition is pretty simple : the complex σ(C) is a subdivision of ...
3
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78 views

When is it possible to extend several linear orders defined “locally” into a single linear order defined “globally”?

This is a somewhat fuzzy question, so I will try my best to give a formulation which includes everything relevant while excluding everything else. I would like to find out if anyone else has studied ...
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67 views

Can a spherical simplicial complex have more than one “central” inversion?

Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if $\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and $\phi$ is not ...
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107 views

“Baues poset” of shellings of simplicial polytope?

Let me start with some background I want to use as analogy. Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: ...
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34 views

Are there structural properties of minimal torsion parts in simplicial complexes?

Are there any structural criteria to find torsions in a simplicial complex? Are there some sufficient and/or necessary properties of torsion-free simplicial complexes? Would it becomes easier to ...
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56 views

Generalization of Moore graphs — not geometries

I was thinking about the following problem: Suppose you have a $k$-uniform hyper graph (simplicial complex of dimension $k$) with a complete $(k-1)$-skeleton, and some form of regularity (e.g. every ...
7
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2answers
122 views

Constructing a $0/1$ polytope from an abstract simplicial complex

Let us fix $\Delta$ a finite simplicial complex, and label the vertices of $\Delta$ as $\{1,2,\ldots,n\}$. For each $F\in \Delta$ let us consider the point in $\mathbb{R}^n$ given by: $$e_F := \sum_{i\...
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2answers
149 views

Inequality of $h$-vectors of shellable simplicial complexes

I've been studying the article of Bjorner entitled "Homology and shellability of matroid complexes". At a certain point he states an exercise that says: Let $\Delta$ be a shellable ...
3
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3answers
289 views

Are there characteristic-dependent Betti numbers in characteristic not equal to two?

Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
3
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0answers
120 views

Skeleton of $\mathcal{G}$-simplicial complex

I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition : $\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{...
4
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1answer
209 views

What is the significance of ear decompositions for non-graphic matroids?

On Wikipedia there is subsection in the article on ear decompositions of graphs titled "Matroids": Now as defined above, the circuits of a matroid can not always be listed to satisfy the ...
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141 views

Is there a combinatorial representation of general topological manifolds similar to triangulations?

Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
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2answers
257 views

Does the union of the subcomplexes of $X$ that contain a given subcomplex and whose inclusion in $X$ is trivial on $\pi_1$, have trivial $\pi_1$?

Let $X$ be a simplicial complex and let $A \subset X$ be a contractible subcomplex on the same set of vertices as $X$. Is it true that the union $$\bigcup C$$ taken over all complexes $A \subset C \...
5
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1answer
115 views

Existence of equivariant triangulations

I'm looking for a compatibility result which links two types of structures that could be imposed on a topological space $X$: Call $X$ triangulable if there exists a finite simplicial complex $K$ ...
11
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1answer
198 views

A class of simplicial complexes defined by arithmetic properties

The purpose of my question is to ask about properties in a certain class of 3-dimensional (and other odd dimensional) simplicial complexes. I will first describe the construction in 3 dimension and ...
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103 views

On a generalization of the Borsuk-Ulam theorem / Tucker's lemma for a map from simplex to its boundary

The Borsuk-Ulam theorem is equivalent to $S^{n-1}$ not being a retract of $B^n$. <totally wrong! or else 2+2=4 is equivalent to the Poincare conjecture/thm> How shall i prove the following ...
3
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90 views

F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex

Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex? Thank you
5
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1answer
167 views

Visualize (co)sketeton of a simplicial set (geometrical intuition)

I want to understand if there is an intuition approchable with most possible 'elementary geometrical' knowledge for $n$-(co)skeleta of simplicial sets? Formally sketleton & coskeleton functions ...
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102 views

Combinatorial condition for orientability on simplicial complexes

Let $K$ be a simplicial complex whose geometric realization is a topological or smooth manifold. Is it possible to restate the condition of orientability of $M$ exclusively in (combinatorial) terms of ...
2
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3answers
355 views

Alternating sum over collections of sets

Let $\mathbf{P}$ be a collection of subsets of a finite set $X$. Let $\mathscr{S}$ be the set of all subsets $\mathbf{S}\subset \mathbf{P}$ such that $\bigcup_{S\in \mathbf{S}} S = X$. Can one give a ...
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124 views

Acyclic extensions of acyclic simplicial complexes

Say an abstract simplicial complex $X$ is acyclic if its reduced integral simplicial homology groups $\tilde{\mathrm{H}}^{\Delta}_p(X)$ vanish for all $p\geq 0$. Is it the case that, for any $n>0$, ...
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2answers
361 views

Alternating sum over collections closed under containment

Let $\mathscr{C}$ be a collection of subsets of a finite set $P$. Assume $\mathscr{C}$ is closed under containment: if $S\subset P$ is in $\mathscr{C}$, then every set $S'\subset P$ containing $S$ is ...
6
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1answer
98 views

Subdivision of closed homology manifold reference request

I am interested in the barycentric subdivision of closed homology manifolds. Definition A (finite) simplicial complex $K$ is a closed homology manifold of dimension $n$ if for every $k$-simplex, its ...
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110 views

Small flag triangulations

In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...
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128 views

How much smaller is the Čech complex than the Vietoris-Rips complex?

The Čech complex is a subcomplex of the Vietoris-Rips complex. The V-R complex includes as a simplex a set of points with pairwise distances at most $\epsilon$, whereas the Č complex includes as a ...
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0answers
65 views

Graphs with vanishing homology and behaviour of the suspension of that graphs

Let $G$ be a simple graph. The clique complex, $\Delta(G)$ of the graph $G$ is the simplicial complex given by the collection of all complete subgraph of $G.$ Now we define the graph homology $H^{Gr}_\...
2
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1answer
189 views

Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...
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55 views

If all 2-faces of a polytope are $2n$-gons, is the edge-graph bipartite?

This question on MSE has not received a satisfying answer. It can be summarized as follows: Question: Is is true that the edge-graph of a (convex) polytope is bipartite if and only if all 2-faces ...
9
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1answer
258 views

Simplicial simple homotopy vs. cellular simple homotopy

I recently started reading up on simple homotopy theory. Here is a question I stumbled upon. In his 1938 Paper Simplicial Spaces, Nuclei and m-Groups Whitehead introduced the notion of elementary ...
2
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1answer
69 views

Uniquely describing a polytopal complex by prescribing the local structure around its vertices

Let $C$ be a $d$-dimensional (abstract) polytopal complex. Most of what I say below could be asked in this general setting, but for a start, let's further restrict to simple polytopal spheres, that is,...
2
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1answer
127 views

simplicial complex of two covers

Given two covers $\{U_a,U_b,\dots\}$ and $\{V_1,V_2,\dots\}$ of a space $X$, what is the appropriate idea of simplicial complex? As far as I see there are two ideas, and I was wondering where these ...
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3answers
87 views

Examples for simplicial complexes in which every k-edge is contained in exactly $d$ $k+1$-edges

Are there any(other than the full complex/1-case)? Is there a name for this ($k$-edge-regular I call it)? Thanks.
2
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1answer
94 views

What is a sufficient set of links in a simplicial complex to represent any PL manifold?

The link of a vertex in a $n$-dimensional simplicial complex is the $(n-1)$-dimensional simplicial complex formed by the $(n-1)$-simplices that together with the vertex span a $n$-simplex. A ...
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4answers
226 views

Generalization of independence complex of graphs

Let $G$ be an undirected graph with no multiple edges or loops. Recall that the independece system $\mathcal{I}(G)$ consists of all those subsets $A$ of the vertex set such that the induced subgraph $...
7
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106 views

Cycles in Tits building

Tits building for an $n$-dimensional vector space $V$ is defined to be the simplicial complex corresponding to the poset of proper and non-zero subspaces of $V$. It is denoted by $T(V)$. This is known ...
2
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1answer
223 views

Converse of Sperner's lemma

The famous Sperner's lemma states that, if a labeling of a triangulation of a simplex satisfies certain conditions on the boundary, then there must exist a sub-simplex in which all labels are ...
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0answers
41 views

Smooth subdivision which is not rectilinear

What is the simplest example of a smooth subdivision of the standard simplex $\Delta^n$ which can not be realized as a rectilinear subdivision? That is I want a simplicial complex $K$ for which there ...
8
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1answer
157 views

Extending a triangulation of the boundary of $M \times I$

(Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $M$, I mean a homeomorphism with the geometric realization of a simplicial complex, $h: |K| \...
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0answers
101 views

simplicial nomenclature and homology

Suppose I have a simplicial complex $K$ constructed by taking two simplicial complexes $K_1$ and $K_2,$ and coning off ever vertex of $K_1$ to all of $K_2$ and vice versa (so, a direct generalization ...
7
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1answer
265 views

Faithfully flat descent for modules from the simplicial point of view

Let $R \rightarrow R'$ be a faithfully flat ring map, let $M$ be an $R$-module, and let $M_n$ be the base change of $M$ to the tensor product of $n + 1$ copies of $R'$ over $R$. One way to formulate ...
5
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1answer
112 views

Sufficient criterion for a simplicial sphere to be polytopal

Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)? ...
8
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1answer
348 views

Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...
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2answers
310 views

How to compare different type of simplicial complex?

I know different type of simplicial complexes like Rips, Alpha, witness, etc. I like to know more about if we have a point cloud which one of them should I use? How do we compare their performance on ...
4
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0answers
145 views

Are triangulations with common refinements PL-homeomorphic?

Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
6
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1answer
357 views

Are triangulations of compact manifolds PL homeomorphic?

I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes ...
12
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0answers
117 views

Finite list of neighborhoods to approximate any finite simplicial complex

It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...