# Questions tagged [simplicial-complex]

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.

470 questions
Filter by
Sorted by
Tagged with
28 views

### Simplicial complex

I started to learn about "simplicial complex" and read about applications but it was very difficult for me to understand these applications, my question is as below what is the importance ...
22 views

### The norm in the n-simplex

İ have this question and my attempt to solve it as the following : Consider an n-simplex $[p_0, p_1,...,p_n]$. If $a,b \in [p_0, > p_1,...,p_n]$, then show that $||a − b|| \leq \sup_i ||a − p_i||$....
17 views

### Cellular Approximation for Homeomorphism on a Compact Surface.

In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows: Suppose $M$ is a ...
1 vote
30 views

### Can a cube be decomposed into tetrahedrons for any configuration of diagonals?

Suppose we have a cube and configuration of diagonals of the faces of this cube. Is there a way to separate this cube into tetrahedrons, such that for each edge of the each tetrahedron, if the edge ...
1 vote
21 views

### Example of connectivity of join is strictly greater than the sum of connectivities plus 2?

Given a topological space $X$, $X$ is $k$-connected if any $-1\le \ell \le k$ and continuous map $f:S^{\ell}\to X$ can be extended to $\bar{f}:B^{\ell+1}\to X$, where $S^\ell$ is viewed as the ...
46 views

### Why is this example (non-simplicial $\mathbb R$-tree) not a simplicial complex?

This is an example of a non-simplicial $\mathbb R$-tree (from Wikipedia): Start with the interval $[0,2]$, for each positive integer $n$, glue an interval of length $1$ to the point $1-1/n$ in the ...
68 views

### Set of 0-cells of a simplicial complex and discreteness

When given a simplicial complex, does its set of $0$-cells have to be a discrete set? In particular, can the set $\{0, 1/n\}_{n=1,...}$ be a simplicial complex (consisting of $0$-cells)? From the ...
8 views

### On the definition of Chow rings for atomic lattices

In the following paper https://arxiv.org/pdf/math/0305142.pdf the authors introduce an algebra $D(\mathcal{L},\mathcal{G})$ (see Definition 3). However, they assert that "although D is defined ...
1 vote
31 views

### Computing singular homology of a cylinder with a bottom, using Mayer Vietoris Sequence

I am trying to practice using MVS on an easy example, a cylinder with a bottom. Explicitly, something like $S^1 \times [0,1]$ with a copy of $D^2$ glued at one end. Call this object $X$. Then I ...
23 views

27 views

### Can a single simplex be a boundary of a higher dimensional chain?

I know that the boundary of a single $d-$dimensional simplex would be a chain of $d+1$, $d-1$-dimensional faces. However, is it possible for a single $d-1$-dimensional simplex to be a boundary of a ...
45 views

### Criterion for the inverse map to be simplicial

Let $K$ and $L$ be abstract simplicial complexes and let $V(K)$, $V(L)$ denote their vertex sets. Then a simplicial map $K \to L$ is a map $f\colon V(K)\to V(L)$ such that $\{v_0,\dots, v_n\}\in K$ ...
32 views

### Extending simplicial maps between filtrations to homology groups.

Suppose I have 2 filtrations of simplicial complexes $K,G$: $\{K_{\alpha}\}_{\alpha\in\mathbb{R}},\{G_{\beta}\}_{\beta\in\mathbb{R}}$. Here, $K_{\alpha}$ is a subcomplex of $K$ and if $\alpha\leq\beta$...
39 views

### Triangulation of 3-sphere and Hopf Fibration

I am currently reading the paper A Minimal Triangulation of the Hopf Map and its Application. In the paper, the authors are trying to describe a triangulation of the 3-sphere into a (abstract) ...
92 views

### On simplicial complexes and their geometric realization

Simplicial complexes can be defined in two different way, i.e. either abstractly as purely combinatorial objects, or embedded in Euclidean space. Let me briefly mention which definitions I use exactly:...
26 views

### Binomial coefficient identities from h-vector of a simplicity complex. [duplicate]

My question is to show the below equality $$\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\binom{d-j}{i}=0$$ when $d>i \geq 0$ for any integers $d,i$. This inequality is came from Stanley's note. Given an $f$-...
57 views

### How to show the exactness of the simplicial complex of free abelian groups generated by configurations of a finite set?

Given a finite set $X$ with $|X|=N>2$, we can construct a simplicial free abelian group $C_*(X)$ (which is a chain complex) defined as follows: for each $n\geq0$, $C_n$ is defined to be the free ...
57 views

### Every compact subset can intersect only finitely many simplices

Well my question is almost in the title: how can I prove that every compact subset C of |K| can intersect only finitely many simplices, where K is a infinite simplicial complex? In particular given C ...
1 vote
81 views

### What is the homology of the 1, 2 and 3 simplex?

Here is the question I am trying to imagine and solve: Compute the homology groups of the $\Delta$-simplex $X$ obtained from $\Delta^n$ by identifying all faces of the same dimension. Thus $X$ has a ...
45 views

22 views

### Classify surface given by $abca^{-1}b^{-1}c^{-1}$

I'ven been solving problems from my Topology course, and don't understand something I saw while reading my solved examples. Here's a problem that will let me show my point: Given $X$ a compact ...
1 vote
25 views

### Simplicial Approximation for self maps

Fix a finite simplicial complex $K$ and a continuous map $f:|K| \rightarrow |K|$. The simplicial approximation theorem as usually stated guarantees that $\exists n$ such that $f$ is homotopic to a ...
1 vote
22 views

### Composed relations and simplicial complexes?

Say you have two relations $R$ and $S$ on finite sets such that the composition $RS$ is well-defined. Each of these has a corresponding Dowker complex (actually two, but they are homotopy equivalent). ...
71 views

### 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, ...
25 views

### What is the homological connectivity of the complete abstract simplicial complex?

I am trying to understand the concept of homological connectivity of an abstract simplicial complex. Specifically, I am trying to compute the homological connectivity of the complete abstract ...
1 vote
53 views

### Equivalence of homology theories via acyclic models

I'm looking for a proof of "any two homology theories are equivalent" (obviously with some other hypothesis) via the Acyclic Models Theorem. I know that this is an application of the Acyclic ...
72 views

### Triangulations of the Torus (Example 4 from Munkres' Chapter 1.3)

Below it is possible to find an extract from Chapter 1.3 of Munkres' "Elements of Algebraic Topology", which concerns the triangulation of the torus. I have the following question regarding ...
39 views

26 views

### Why is the following a cubical complex

The german Wikipedia-page on cubical complexes has the following example for a cubical complex. I don't understand how the 45° rotated square on the right is the product of elementary intervals. As I ...