# Questions tagged [simplicial-complexes]

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168
questions

**9**

votes

**0**answers

100 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 ...

**1**

vote

**0**answers

49 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**

votes

**1**answer

125 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 ...

**4**

votes

**0**answers

43 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 ...

**7**

votes

**1**answer

187 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**

votes

**0**answers

55 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,...

**1**

vote

**1**answer

63 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 ...

**0**

votes

**3**answers

82 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**

votes

**1**answer

73 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 ...

**2**

votes

**2**answers

111 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 $...

**5**

votes

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88 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**

votes

**1**answer

213 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 ...

**1**

vote

**0**answers

38 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 ...

**7**

votes

**1**answer

111 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| \...

**1**

vote

**0**answers

100 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**

votes

**1**answer

247 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**

votes

**1**answer

94 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**

votes

**1**answer

287 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\...

**3**

votes

**2**answers

199 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**

votes

**0**answers

125 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**

votes

**1**answer

333 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**

votes

**0**answers

115 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 $> 3$...

**3**

votes

**1**answer

156 views

### Does homotopy equivalence to a wedge of spheres imply shellability?

It's pretty clear that for a simplicial complex $\Delta$, shellability of the complex implies that it is homotopy equivalent to a wedge of spheres. However, does the converse hold? That is, does $\...

**9**

votes

**1**answer

151 views

### Matroidal simplicial posets?

A simplicial poset is a finite poset $P$ with minimial element $\hat{0}$ such that every interval $[\hat{0},x]$ is isomorphic to a Boolean lattice. Simplicial posets are generalizations of simplicial ...

**13**

votes

**1**answer

419 views

### Turning simplicial complexes into simplicial sets without ordering the vertices

Given an abstract simplicial complex $K$, one can make a simplicial
set $X(K)$ with $n$-simplices given by sequences $(x_0, \ldots, x_n)$
such that $\{x_0, x_1, \ldots, x_n\}$ is a simplex of $K$. The ...

**7**

votes

**0**answers

102 views

### Criteria for a poset complex to be contractible

I would like to know if there are nice criteria to know if the ordered complex $C$ induced by a poset is contractible. I am also interested in the same question for subcomplexes of $C$.
$C$ happens ...

**2**

votes

**0**answers

51 views

### Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes:
Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...

**4**

votes

**1**answer

174 views

### Subdivision of simplicial sets but not the barycentric one!

Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of `finer' generalising that of 'finer triangulation' of ...

**6**

votes

**0**answers

116 views

### Is there any work in topological data analysis on something like “Voronoi complexes”?

Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...

**16**

votes

**1**answer

778 views

### Can one determine the dimension of a manifold given its 1-skeleton?

This may be an easy question, but I can't think of the answer at hand.
Suppose that I have a triangulated $n$-manifold $M$ (satisfying any set of conditions that you feel like). Suppose that I give ...

**7**

votes

**1**answer

314 views

### Inequality number of facets simplicial complex

In a recent preprint, Adiprasito proves that if $\Delta$ is a simplicial complex of dimension $d$ that can be embdedded in a $2d$-dimensional homology sphere (say $\Sigma$) that satisfies a version of ...

**1**

vote

**1**answer

122 views

### Number of free faces given n 0-simplexes

Here is my question: How to construct a simplicial complex with $n$ 0-simplex which has the maximum number of free faces? Is there any research topic about this? And is there any relationship between ...

**2**

votes

**0**answers

108 views

### Nerve theorem for locally infinite covers by subcomplexes

Let $Y$ be a simplicial complex and let $\{Y_i\}_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup_{i\in I}Y_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each ...

**5**

votes

**1**answer

316 views

### Criterion for acyclicity of flag complexes

Let $\Delta$ be a flag complex on $n$ vertices. Let $r$ be the smallest size of the facets of $\Delta$. Suppose that $2r>n$. Must $\Delta$ be acyclic?

**11**

votes

**0**answers

443 views

### Is there an expository account of homology of simplicial sets that does not assume prior familiarity with any variant of homology?

There are numerous expositions of simplicial homology in the literature.
Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes.
Hatcher in “Algebraic ...

**8**

votes

**2**answers

428 views

### Simplicial set are to cubical sets what simplicial complexes are to …?

Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the ...

**3**

votes

**1**answer

179 views

### Change of Betti numbers under simplicial maps

Let $\Delta$ be a simplicial complex on $n$ vertices, and $\phi$ a simplicial map that identifies two vertices $x$ and $y$ of $\Delta$. I want to show that the Betti numbers of $\phi(\Delta)$ cannot ...

**2**

votes

**0**answers

85 views

### Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence
$$
1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1.
$$
The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...

**5**

votes

**0**answers

188 views

### Can we represent partitions by mutually parallel lines in the plane?

Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...

**9**

votes

**1**answer

324 views

### Closed orientable surfaces have even Euler characteristic

It is of course completely standard that closed orientable surfaces have even Euler characteristic. What is the most elementary proof of this?
More specifically, suppose I have a finite simplicial ...

**2**

votes

**1**answer

284 views

### How does the high-dimensional combinatorial Laplacian work?

When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,...

**14**

votes

**0**answers

196 views

### How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$

Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...

**8**

votes

**1**answer

306 views

### Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...

**7**

votes

**2**answers

597 views

### Any 3-manifold can be realized as the boundary of a 4-manifold

We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...

**3**

votes

**0**answers

74 views

### Have partition functions of abstract simplicial complexes been examined?

Many complicated probability distributions arising in electrical engineering and machine learning have a simple expression as a sum of products that can also be encoded in a factor graph. The ...

**4**

votes

**1**answer

178 views

### Betti numbers of a Cohen-Macaulay Module in small projective dimension

I am trying to compute the Betti numbers of some Stanley-Reisner ring $R_\Delta$, where the underlying complex $\Delta$ is shellable and the projective dimension of the $R_\Delta$ is $3\text{ or }4$. ...

**4**

votes

**1**answer

85 views

### Simplicial Pseudomanifolds with Boundary - Bounding number of maximal faces in terms of number of vertices and dimension

A simplicial pseudo-manifold of dimension $d$ with boundary is a simplicial complex satisfying the following conditions.
Every maximal face has dimension $d$
Each face of dimension $d-1$ is a face ...

**10**

votes

**3**answers

473 views

### Is there a discrete lattice analogue of conformal transformations?

There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...

**8**

votes

**2**answers

247 views

### Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely
\begin{equation}
\chi=\...

**17**

votes

**3**answers

406 views

### Vietoris-Rips complex homology of a higher degree than the ambient dimension

Assuming we have a set of points $X=\{x_1,..,x_n\}$, all in $\mathbb{R}^d$, and construct the Vietoris-Rips-Complex $V_\epsilon (X)$ for some distance parameter $\epsilon > 0$.
Is it possible to ...