The simplicial-complexes tag has no wiki summary.

**4**

votes

**2**answers

149 views

### Simplicial complices on unlabelled vertices

My question is about (abstract) simplicial complices.
In particular, how many are they if I consider $n$ unlabelled vertices?
For example, if $n=4$, the two complices
$$
\{\varnothing, \{1\}, \{2\}, ...

**3**

votes

**1**answer

160 views

### Kan condition in simplicial homotopy theory

I know Kan condition (see http://en.wikipedia.org/wiki/Kan_fibration) is something like homotopy extension condition, and I know this condition ensures homotopy defined by the naive idea to be an ...

**12**

votes

**0**answers

127 views

### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...

**8**

votes

**0**answers

161 views

### Consequences of Zeeman's conjecture

Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
Zeeman showed that this implies the Poincaré conjecture in ...

**2**

votes

**0**answers

53 views

### Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?

Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | ...

**2**

votes

**0**answers

79 views

### A natural simplicial object in the simplicial category (?)

In several works (es. [CS]) the study of the properties of the simplicial category $\Delta$ reveals fundamental aspects of universal properties (eg monoid generator) or basic constructions (eg ...

**2**

votes

**2**answers

131 views

### Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.
Now, if $G$ is a well-covered graph (where all maximal ...

**14**

votes

**1**answer

1k views

### What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...)
Are there examples of results in "classical" [*] graph theory that have
been achieved by using simplicial ...

**5**

votes

**0**answers

151 views

### Is Euler-characteristic of a simplicial complex on $n$ vertices and $f$ facets at most $n^{O(\log f)}$?

(Definition: Facet = Maximal Face)
This question is a continuation of the previous one that I had asked a couple of years ago: Is Euler characteristic of a simplicial complex upper bounded by a ...

**11**

votes

**0**answers

404 views

### Has this chain complex associated with a simplicial complex been studied before?

I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference.
Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...

**5**

votes

**1**answer

108 views

### References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other.
This ...

**3**

votes

**1**answer

75 views

### Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...

**1**

vote

**0**answers

119 views

### fixed simplicial complex under group action

I have found in an article dealing with combinatorial manifolds the following definition:
Let $C$ be a finite simplicial complex, and let $G$ be a finite group acting by automorphisms of $C$. The ...

**2**

votes

**0**answers

211 views

### When is the realisation of a simplicial set a manifold?

It is known that a simplicial complex is homeomorphic to a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dimension, the ...

**1**

vote

**1**answer

175 views

### Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...

**3**

votes

**1**answer

267 views

### What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem:
Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...

**-1**

votes

**1**answer

74 views

### Poset complex of reverse ordering [closed]

This might be too easy but I cannot proof it easily. Any reference or hint will be great.
Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...

**0**

votes

**0**answers

176 views

### Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims:
Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...

**1**

vote

**1**answer

162 views

### 3-complexes not embeddable in 3-space

My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?

**5**

votes

**1**answer

129 views

### Injective simplicial maps between Arc complexes

Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...

**3**

votes

**1**answer

144 views

### Lipschitz Approximation to a PW Smooth Map

Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...

**3**

votes

**1**answer

132 views

### Thinking about the quadratic dual graphically

Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed ...

**1**

vote

**1**answer

184 views

### simplicial complex equipped with barycenric metric is complete [closed]

Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a ...

**10**

votes

**2**answers

481 views

### How many triangulations of the genus $g$ surface on $n$ vertices?

By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations $t(n)$ of the ...

**1**

vote

**0**answers

88 views

### Is this basis of simplex polynomials known?

Put $R_n=\mathbb{R}[t_0,\dotsc,t_n]/(\sum_it_i-1)$ (the ring of polynomial functions on the $n$-simplex). Consider a monomial $t^a=t_0^{a_0}\dotsb t_n^{a_n}$. Let $(b_0,\dotsc,b_n)$ be the sequence ...

**5**

votes

**1**answer

279 views

### Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...

**2**

votes

**2**answers

403 views

### Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...

**3**

votes

**0**answers

198 views

### Simplicial chain complex with ordered simplices

Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols ...

**7**

votes

**2**answers

378 views

### Combinatorial distance between simplicial complexes

Let $K_1$ and $K_2$ be two simplicial complexes.
I am seeking a measure of the distance between $K_1$ and $K_2$ when
viewed as combinatorial objects.
What I have in mind is something like this.
...

**1**

vote

**0**answers

138 views

### Cohomology with compact support and the nerve of a recouvrement

Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...

**3**

votes

**1**answer

183 views

### Are there “geometrically nice” sets from which to construct coverings that admit “Vietoris-Rips like” approximations to the nerve?

It is well known that the nerve (or Čech complex) of a covering consisting of metric balls with a common fixed radius is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its ...

**7**

votes

**2**answers

249 views

### Higher-dimensional Fáry's theorem?

Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges.
For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex ...

**24**

votes

**3**answers

1k views

### Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ ...

**4**

votes

**0**answers

156 views

### When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where ...

**17**

votes

**2**answers

477 views

### Is there an analogue of the Erdős–Gallai theorem for simplicial complexes?

The Erdős–Gallai theorem gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph.
In particular $d_1 \ge d_2 \ge \dots \ge d_n$ ...

**46**

votes

**0**answers

3k views

### The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...

**2**

votes

**1**answer

133 views

### coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...

**14**

votes

**1**answer

533 views

### Homotopy of random simplicial complexes

A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with ...

**12**

votes

**1**answer

390 views

### Simply connected simplicial complexes

Let $Z$ be a simply connected, two dimensional simplicial complex.
Let $X\subset Z$ be a finite subcomplex with nontrivial $\pi_{1}$.
Must there exist a finite, simply connected subcomplex $Y\subset ...

**11**

votes

**3**answers

522 views

### Effect on homology of decorating vertices of a simplicial complex

In my research, the following construction came up.
Let $X$ be an $n$-dimensional simplicial complex. For an integer $m \geq 1$, let $X[m]$ denote the following simplicial complex. The vertices of ...

**1**

vote

**0**answers

95 views

### Constructible fans

Hi,
the face poset of a complete simplicial fan is a simplicial sphere. It is not shellable in general, but Kleinschmidt, Smilansky and Onn shown that it is always partitionnable. Constructibility is ...

**1**

vote

**0**answers

103 views

### Do bistellar flips in a simplicial sphere preserve facet connectivity?

Define two facets of a simplicial sphere as k-connected if there are at least k disjoint edge-vertex paths between them in their facet-ridge graphs.
Do bistellar flips preserve facet k-connectivity?
...

**15**

votes

**3**answers

978 views

### Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?

What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or ...

**12**

votes

**2**answers

761 views

### A simplicial complex which is not collapsible, but whose barycentric subdivision is

Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the ...

**23**

votes

**0**answers

1k views

### Are there lots of integer homology three-spheres?

The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and ...

**9**

votes

**4**answers

422 views

### Can one do without a classifying space when showing vanishing of cohomology

Let $G$ be a discrete group and $A$ an abelian group, then $H^n (G,A)$ can be defined as $$ H^n (G,A) = H^n (B_G, A)$$
Where $B_G$ is the classifying space of $G$, i.e. $B_G = E_G / G$ where $E_G$ is ...

**7**

votes

**2**answers

485 views

### Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!) What can be said about subcomplexes of 2-complexes deformation retractible onto graphs.

I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
Is it true that the Euler characteristic of a finite connected aspherical ...

**0**

votes

**0**answers

214 views

### Simple question on the foundations of spin foam formalism

To make it simple, take the spin foam formalism of ($SU(2)$) 3D gravity. My question is about the choice of the data that will replace the (smoothly defined) fields $e$ (the triad) and $\omega$ (the ...

**1**

vote

**1**answer

450 views

### “Face” numbers for tropical Grassmannian G′_2,7 simplical complex ?

In an excerpt of an article by Bernd Sturmfels, I found:
Theorem 5.5. The tropical Grassmannian $G^{′}_{2,n}$ is a simplical complex known as the space of phylogenetic trees.... It is denoted by ...

**2**

votes

**1**answer

504 views

### High connectivity arguments for simplicial complexes

I am wondering if there are "elementary" ways to check for $n$-connectedness of simplicial complexes in terms of simple conditions on vertices.
For example
Let $X$ be the curve complex on a surface ...