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.
511
questions
1
vote
1
answer
46
views
Non-metrizability of specific polyhedron by showing that first countability axiom fails (Munkres ETA, exercise 2.4, p. 14)
Let $K$ be a collection of one simplices $\sigma_1,\sigma_2,\ldots$ in $\mathbb{R}^2$ where each vertex $\sigma_i$ has vertices $(0,0)$ and $(1,1/i)$. This collection $K$ of $1$-simplicies constutes a ...
- 211
1
vote
0
answers
33
views
Let K be a simplicial complex and σ a simplex of K. Show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty.
I wanna know if I choose the right way to do my proof? it's correct?
To show that the link lk(σ) is a subcomplex of K, if lk(σ) is not empty, we need to show two things:
Every simplex of lk(σ) is a ...
- 11
0
votes
0
answers
21
views
Triangulation of a real projective plane that has a non contractible loop
In A.Costa and M.Farber's 2015 article on random simplicial complexes, they write:
Let $Z = P^2 \cup D^2$ be the following 2-complex. Here $P^2$ is a
triangulated real projective plane having a cycle $...
- 321
1
vote
1
answer
62
views
Is there a definition of the "boundary" of a simplicial complex?
Let $S$ be the abstract simplicial complex with facets $\{A,B,D\}$, $\{A,C,D\}$, $\{B,C,D\}$. Its geometric representation is homeomorphic to a disc:
In this picture, the geometric representation of ...
1
vote
0
answers
42
views
Proving an Abstract Simplicial Complex has Geometric Realization
The following is an exercise from Lee's Introduction to Topological Manifolds:
Show that an abstract simplicial complex is the vertex scheme of a
Euclidean complex if and only if it is finite-...
- 1,985
1
vote
2
answers
37
views
Can there be a simplicial map from a triangulation of the disc to a triangulation of the circle?
A triangulation of a geometric object $G$ is an abstract simplicial complex whose gemoetric realization is homeomorphic to $G$. For example, the abstract complex with facets {A,B},{B,C},{C,A} is a ...
0
votes
0
answers
38
views
Continuity of Simplicial Maps
John Lee's Introduction to Topological Manifolds gives the following definition for a simplicial map:
"Suppose $K$ and $L$ are simplicial complexes [in $\mathbb{R}^n$]...A simplicial map from $K$ ...
- 1,985
0
votes
0
answers
14
views
Asymptotic for number of simplices in the barycentric subdivision of a Rips complex
This is related to a research problem I am working on.
Let $Rips(X)_r$ denote the Vietoris-Rips complex of a point set $X$ ($\vert X \vert = n$) in a metric space, at some scale parameter $r$.
It is ...
- 1,831
0
votes
1
answer
59
views
Big-$O$ size of the double sum $\sum_{k=0}^{n} \left( \sum_{i=1}^{k+2} (-1)^{k+i} {k+1\choose i-1} i^{n+1} \right)$
I am trying to determine the big-$O$ size of the following double sum. I am pretty sure it is exponential in the size of $n$ (i.e. $O(c^n)$ for some constant $c$), but I would like to know how to ...
- 1,831
2
votes
1
answer
72
views
Is every $\Delta$-complex realizable as Simplicial complex, ie triagulazable?
In the following I will use definitions and
constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets.
I'm ...
2
votes
1
answer
41
views
Does there exist a class of graphs with $O(n^2)$ maximal cliques which can be realized as the 1-skeleton of a Vietoris-Rips complex in $\mathbb R^2$?
Let $\mathfrak G$ be a class of graphs with quadratically many maximal cliques in the size of the vertex set. In other words, if $G=(V,E)\in \mathfrak G$ with $|V|=n$, the graph $G$ contains $O(n^2)$ ...
- 1,831
0
votes
2
answers
48
views
"Compatible /admissible" maps of $\Delta$-complexes
In the following I'm going to use definitions and
constructions of simplicial complexes and $\Delta$-complexes/sets from Greg Friedman's An elementary illustrated Introduction to simplicial sets.
...
1
vote
0
answers
42
views
Homotypy types of Simplicial Complexes vs $\Delta$-Complexes vs CW Complexes [duplicate]
I want to compare diferences between homotopy types of
simplicial complexes, $\Delta$-complexes and CW-complexes.
(to avoid confusions I'm using the definitions, notations und
constructions of ...
0
votes
0
answers
63
views
Is there algebra (algebraic manipulation) of simplicial sets?
Is there algebra of simplicial sets? For example, symbolic representation of simplicial sets and operations on those representations that allow to construct new simplicial set from existing one – join,...
- 1,353
1
vote
0
answers
44
views
Is the Hodge decomposition on a simplicial complex canonical?
Motivation
The question arises because in the paper arXiv:1005.2405 the authors pull back the Hodge decomposition on a simplicial complex from $C^1$ to $(C^0)^N$ for some integer $N$ via some ...
- 384
4
votes
1
answer
117
views
Do isomorphic simplicial chain complexes give isomorphic simplicial complexes?
It seems that chain complexes have all information of simplicial complexes.
If we have an isomorphism between chain complexes which are induced by simplicial complexes, can we conclude that two ...
- 51
1
vote
1
answer
50
views
Obstruction for a compact simplicial complex
$\mathbf {The \ Problem \ is}:$ Let $X$ be a compact simplicial complex and $Y$ be based, connected space with $f:X\to Y.$ If $X$ is simply connected and $f_k:=f\mid_{X^k}$. Show that $f_1$ is ...
- 2,037
0
votes
0
answers
10
views
Orientation of simplicial complexes [duplicate]
I'm trying to understand simplicial homology over $\mathbb Q$ and I'm having some problems in grasping with the definition of orientation of an abstract simplicial complex, the major one being that I ...
- 486
0
votes
1
answer
49
views
A combinatorial definition of a piecewise-linear sphere
Define a piecewise-linear sphere (PL sphere) as an abstract simplicial complex, whose gemoetric realization is homeomorphic to a sphere. Examples:
The complex over {1,2,3} which contains the sets {1,...
1
vote
1
answer
21
views
An extension of PL triangulation of $S^k$
Let $\mathcal{T}$ be a piecewise linear triangulation of the sphere $S^k$. I am wondering is it possible to get an extension of $\mathcal{T}$ to a piecewise linear triangulation $\mathcal{T}'$ of the ...
- 1,717
1
vote
1
answer
29
views
Smallest offset so that the Rips complex is always a subcomplex of the Cech complex.
Let $X$ be a finite set of points in $\mathbb R^n$.
For $\epsilon >0$, let $\mathcal R_\epsilon (X)$ and $\mathcal C_\epsilon (X)$ be the Rips and Cech complexes on $X$, respectively.
It's well-...
- 1,831
2
votes
1
answer
72
views
"singular homology = simplicial homology" relative to a fibration
Everyone learns in the first course in algebraic topology that the singular homology of a topological space with a simplicial decomposition is isomorphic to its simplicial homology.
I want to ask if ...
- 395
4
votes
1
answer
63
views
Can we ignore higher dimensional information when computing the geometric realisation of an $n$-dimensional simplicial set?
$\newcommand{\lan}{\operatorname{Lan}}\newcommand{\tr}{\operatorname{tr}}\newcommand{\H}{\mathsf{H}}\newcommand{\set}{\mathsf{Set}}\newcommand{\T}{\mathsf{Top}}\newcommand{\C}{\mathsf{C}}\newcommand{\...
- 22k
1
vote
0
answers
27
views
converting between simplicial and cell complexes
A cell of an oriented $n$-dimensional simplicial complex consists of the ordered tuple $[v_0, \ldots, v_m]$ of its vertices, modulo positive permutations.
So for example $[v_0, v_1, v_2]$ and $[v_1, ...
- 381
2
votes
1
answer
94
views
How to show there is an $S^4$ included in a simplicial complex?
A $\mathbb{Z}_2$-space is a pair $(T, \nu)$, where $T$ is a topological space and $\nu: T \rightarrow T$, called the $\mathbb{Z}_2$-action, is a
homeomorphism such that $\nu \circ \nu= id_{T}$ . If $(...
- 393
1
vote
1
answer
41
views
Can we take torus like triangulation for a real projective plane RP²?
I study simplicial homology topic in algebraic topology, where I read about the triangulation of compact matritizable spaces and I am going to compute the triangulation of real projective plane RP².
...
- 11
0
votes
0
answers
55
views
Triangulating R^n
I am interested in triangulating $\mathbb{R}^n$ using the standard triangulation of $[0, 1]^n$.
By a triangulation of a subset $X \subseteq \mathbb{R}^n$, I mean a set of $n$-simplices $S$ whose union ...
- 590
1
vote
0
answers
83
views
Why projective plane $\Bbb RP^2$ contains no two cycles?
I am new to simplicial complexes and I am reading a paper on "directed rooted forest in higher dimension" in which in an example it was mentioned that projective plane of dimension 2 contain ...
1
vote
0
answers
50
views
Is image of the cup-product actually a homomorphism?
Given a simplicial complex $K$, its simplicial cochain complex $C^\bullet(K) = C_\bullet(K)^\vee = Hom(C_\bullet(K), \mathbb{Z})$, as well as $p,q \in \mathbb{N}_0$ and $\sigma := \{ \sigma_0, \dots, \...
- 131
0
votes
0
answers
23
views
Projective varieties from Hilbert-Poincaré polynomials
On SINGULAR, the command hilb(); outputs both the first and the second Hilbert-Poincaré polynomials (Q(t) and G(t), respectively), and the dimension and the degree of a corresponding projective ...
- 4,589
0
votes
0
answers
79
views
(Co)homology of partially ordered sets (posets)
I'm having trouble understanding some basic facts on algebraic topology. If I have a topological space $X$ and want to calculate the (co)homology of it, it seems to me to calculate singular (co)...
1
vote
1
answer
25
views
Computing Reduced Homology with SAGEMath
Good morning to everybody. I downloaded SAGEMath9.3 on my Windows in order to make some computations with abstract simplicial complexes.
I found the following reference manual http://www2.math....
- 4,589
0
votes
0
answers
45
views
Can a math software do decide whether an abstract simplicial complex is or not a matroid?
As the title suggests, I need to know if there exists some math software deciding whether an abstract simplicial complex is or not a matroid. In other terms, I would like to give a finite ground set ...
- 4,589
4
votes
0
answers
79
views
Does a sphere always admit a triangulation in which the link of a vertex is a sphere?
In this question, it is asked whether for any triangulation $C$ of a sphere $S^k$, and for any vertex $v$ of $C$, the link of $v$ is homeomorphic to a sphere $S^{k-1}$.
This answer shows a concrete ...
0
votes
0
answers
33
views
Can MACAULAY2 do computations with Simplicial Complexes?
I need to do some computations involving abstract simplicial complexes (non-empty collections of finite subsets closed under taking subsets). More in detail, if I have a simplicial complex, I need to ...
- 4,589
1
vote
1
answer
61
views
Why $\Delta_1(S^1)=\mathbb{Z}$ and $\Delta_n(S^1)=0$ for $n\geq 2$?
I find it difficult to understand it when I read Hatcher's Algebraic Topology. In Example 2.2, I can understand $\Delta_0(S^1)=\mathbb{Z}$. But how to illustrate why $\Delta_1(S^1)=\mathbb{Z}$ and $\...
user893126
0
votes
0
answers
83
views
Understanding triangulable topological spaces
Let $X$ be a topological space. If there exists a simplicial complex $K$ and a homeomorphism $f:|K| \rightarrow X, X$ is said to be triangulable and the pair $(K, f)$ is called a triangulation of $X$.
...
- 9,888
3
votes
1
answer
80
views
Understanding Topology of simplical complexes
Context
I'm trying to understand the Topology of the simplical complexes as explained in this wiki,
First, define $|K|$ as a subset of $[0,1]^{S}$ consisting of functions $t: S \rightarrow[0,1]$ ...
- 9,888
2
votes
1
answer
111
views
Surjection from cohomotopy to cohomology
Given a finite topological simplicial complex $X$ of dimension $m$. There exists a canonical way to define a homomorphism from the n-th cohomotopy group of $X$ to the n-th cohomology group.
I would ...
- 71
0
votes
0
answers
34
views
How to justify this step in proof of Brouwer lemma / about simplices?
I am reading a proof of the fact that sphere doesn't retract to a ball (equivalent to Brouwer's fixed point theorem). The proof is based on Sperner's Lemma. I am stuck at a part of a proof that is in ...
- 1,552
2
votes
0
answers
66
views
Homotopy-theoretic computation of Euler characteristic of skeleton of simplex
The Euler Characteristic of the $k$th skeleton of the $n$-simplex is
$$\chi(\mathrm{sk}_k \Delta^n) = \binom{n + 1}{1} - \binom{n + 1}{2} + \cdots + (-1)^{k + 1}\binom{n + 1}{k + 1} = 1 + (-1)^k\binom{...
user630227
3
votes
1
answer
81
views
Is there a uniform definition of oriented simplex that gives two orientations in all dimensions?
Define a simplex to be a finite subset of vertices: a subset with cardinality $n$ defines an $(n - 1)$-simplex. An oriented simplex is typically defined to be a simplex and an equivalence class of ...
- 33
4
votes
0
answers
43
views
Formula for $r$-graph homology
I ran across the following formula on Wikipedia. If $G = (V,E)$ is a graph, then viewing the graph as a simplicial complex (vertices are $0$-cells and edges are $1$-cells), we have the formula
$$H_1(G)...
- 1,700
2
votes
1
answer
122
views
An equivalent condition of being $d$-Leray ($d$th and above homology groups vanish)
Given a simplicial complex $X$ on vertex set $V$, $X$ is called $d$-Leray if the reduced homology group satisfying $\tilde{H}_i(Y)=0$ for all induced subcomplexes $Y\subseteq X$ (i.e., Y=X[S] for some ...
- 1,717
1
vote
0
answers
31
views
Euler characteristic of the join of a simplex and the boundary of another simplex
Let $\Delta^n$ be the abstract simplicial complex on $n + 1$ vertices where every nonempty subset of vertices is a face. Let $\partial \Delta^n$ be the abstract simplicial complex on $n + 1$ vertices ...
user630227
0
votes
1
answer
57
views
Barcodes Decomposition of Persistent Homology
Does anyone know if the barcode decomposition of a simplex-wise filtration a multiset? More specifically, can we have multiple barcodes with the same birth time? When I read the paper by Gunnar ...
- 2,772
1
vote
0
answers
64
views
Can higher dimension simplex occur in a triangulation of a $k$-sphere?
A $k$-simplex is a convex hull of $k+1$ points (which are called vertices) in general position in $\mathbb{R}^n$ (for $k\le n$).
A face of a simplex is a simplex spanned by a subset of its vertex set.
...
- 1,717
2
votes
1
answer
136
views
link of a vertex in the triangulation of $S^k$ is a triangulation of $S^{k-1}$?
Let $C$ be a triangulation of sphere $S^k$, i.e., $C$ is a geometric simplicial complex and the union of all simplices in $C$ is homeomorphic to $S^k$.
For a vertex $v$ of $C$, $$link_C(v):=\{\sigma\...
- 1,717
0
votes
1
answer
72
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 $||a − b|| \leq \sup_i ||a − p_i||$....
- 21
0
votes
0
answers
36
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 ...
- 160