# Questions tagged [simplicial-complexes]

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196
questions

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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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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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91 views

### 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 ...

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**1**answer

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 $\mathbf S^d\subset\Bbb R^{d+1}$.
...

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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 ...

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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 ...

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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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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 ...

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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 $\...

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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{...

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**1**answer

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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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 \...

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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 $K$ ...

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votes

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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 ...

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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

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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 ...

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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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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 ...

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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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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}_\...

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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 ...

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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 ...

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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,...

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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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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.

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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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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 $...

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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 ...

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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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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 ...

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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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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 ...

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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 ...

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### 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)?
...

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### 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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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 ...

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### 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 ...

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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 ...

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### 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 $> 3$...