A finite simplicial complex can be defined as a finite collection $K$ of simplices in $\mathbb{R}^N$ that satisfies the following conditions : (1) Any face of a simplex from $K$ is also in $K$ and (2) The intersection of any two simplices in $K$ is a face of both simplices.

# Questions tagged [simplicial-complex]

471 questions

**4**

votes

**1**answer

### When does a map between abstract simplicial complexes induce a homotopy equivalence on their geometric realisations?

Let $\left(X_i,S_i\right)$ for $i=1,2$ be abstract simplicial complexes where $X_i$ are the vertex sets and $S_i\in\mathcal P\left(X_i\right)$ are the sets of simplices. Let $F\colon X_1\to X_2$ be a map of abstract simplicial complexes (for any…

Chetan Vuppulury

- 1,226
- 7
- 18

**4**

votes

**2**answers

### Tucker's lemma, Borsuk-Ulam, triangulating a ball in *truly* antipodally symmetric fashion

I'm attempting to prove Tucker's Lemma from the Borsuk-Ulam theorem by means of the proof sketched as "immediate" on page 36 of Matoušek's Using the Borsuk-Ulam Theorem. In order to do this, I need an auxiliary result.
Let $\Delta$ denote a…

I.M.J. McInnis

- 63
- 5

**4**

votes

**2**answers

### Is there a triangulation of a closed surface with each vertex incident to $n\ge 7$ triangles?

Fix some $n\ge 7$. Is there a finite 2-dimensional simplicial complex with
each vertex is incident to exactly $n$ triangles, and
the complex forms a closed surface.
If the surface is not closed, then uniform tilings of the hyperbolic plane do the…

M. Winter

- 28,034
- 8
- 43
- 92

**4**

votes

**2**answers

### (elementary?) proof of simplicial homology groups of $\Delta_n$

Using the (trivial) CW-complex structure of $\Delta_n$, I would like to compute the homology groups of $\Delta_n$. It's obvious that (for $k \leq n$) $C_k \simeq \mathbb{Z}^{{n+1 \choose k+1}}$, and that $im \delta_{k+1} \subset ker \delta_k$. It's…

vxnture

- 698
- 3
- 10

**4**

votes

**1**answer

### Can I Approximate a Homeomorphism by Permuting a Grid?

Let $f: [0,1]^2 \to [0,1]^2$ be a homeomorphism of the square. By the n-grid I mean the collection of smaller squares $$[a/2^n,a/2^n + 1/2^n] \times [b/2^n,b/2^n + 1/2^n] \text{ for } a,b = 0,\ldots, 2^{n}-1$$ for some natural number $n$. I would…

Daron

- 8,792
- 1
- 16
- 37

**4**

votes

**0**answers

### Higher cup-1 product of coboundaries is also a coboundary?

In the cohomology or the group cohomology theory, suppose $\mu_1$ and $\mu_2$ are coboundaries of arbitrary dimensions,
$$
\mu_1=\delta \eta_1
$$
$$
\mu_2=\delta \eta_2
$$
where $\eta_1$ and $\eta_2$ are their lower dimensional split…

annie marie cœur

- 3,399
- 2
- 14
- 27

**4**

votes

**1**answer

### Simplicial complex with fundamental group $\mathbb{Z}/3\mathbb{Z}$

Is there any example of a simplicial complex with fundamental group $\mathbb{Z}/3\mathbb{Z}$?
(Preferably a “simple” example with as few vertices as possible.)
So far I learnt about $RP^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$.
Thanks.

yoyostein

- 18,172
- 17
- 56
- 120

**4**

votes

**1**answer

### Understanding the $\operatorname{link}$ of a simplex $\sigma$

I have a question that goes as follows:
Let $K$ be an abstract simplicial complex and $\sigma \in K$. Define the link of $\sigma$ as follows:
$$\operatorname{link}_\sigma(K) := \{\tau \in K | \tau \cap \sigma = \emptyset \, ,\, \tau \cup \sigma \in…

Kocmoc

- 51
- 3

**4**

votes

**1**answer

### The KKM lemma implies Sperner's lemma: a direct proof?

It seems fairly well-known that the KKM lemma of Knaster-Kuratowski-Mazurkiewicz implies Sperner's lemma (definitions below). However, I only know of indirect proofs, for instance that KKM can be used to prove Brouwer's fixed point theorem, and that…

Mark

- 472
- 2
- 11

**4**

votes

**1**answer

### Product of standard simplexes is homeomorphic to a standard simplex

The standard $n$-simplex is the set $$\Delta_n=(x_1,...,x_{n+1})\in \{\mathbb R^{n+1}: \Sigma x_i=1, x_i\geq0\}.$$ I'd like to prove that $\prod_{i=1}^p \Delta_{m_i}$ is homeomorphic to $\Delta_m$, with $m=\sum_{i=1}^p m_i$. I noticed that it is…

rgm

- 765
- 7
- 22

**4**

votes

**1**answer

### Triviality of relative cup product $H^2\times H^2\to H^4$ for spaces embeddable to $\Bbb R^4$

Let $X$ be a triangulation of $[0,1]^4$ and $A$ is a simplicial subcomplex of $X$. I would like to show that the cup product $$H^2(X,A)\times H^2(X,A)\to H^4(X,A)$$ is trivial.
It is realitvely easy to show that $H^4(A)$ is trivial, which was…

Peter Franek

- 10,750
- 4
- 24
- 45

**4**

votes

**0**answers

### Sheaves on a Simplicial Complex

I came across a video the other day which discussed practical applications of sheaves. However, rather than defining a sheaf on the open sets of a traditional topological space, the lecturer outlined how you would define them on a simplicial…

Tac-Tics

- 2,013
- 1
- 13
- 28

**4**

votes

**1**answer

### Simplicial Homology Does Not Depend on the Orientation

Let $K$ be a simplicial complex and denote by $K_1$ and $K_2$ the complexes obteined from $K$ with two different orientations. I want to prove that the simplicial homology groups of $K_1$ and $K_2$ are isomorphic, that is, for each $n$ we have $$…

Marco Armenta

- 902
- 5
- 12

**4**

votes

**1**answer

### What is an n-cell?

In my Topology lecture notes, 'n-cell' seems to be mentioned a lot, but it never says what exactly it means. Does it mean $n$-dimensional space?

BetaY

- 593
- 3
- 12

**4**

votes

**1**answer

### simplicial approximation and infinite complexes

It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$.
Does anyone know an example of a map $f$ where $X$ is infinite…

pw1822

- 688
- 3
- 14