# 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.

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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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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}$and$\... 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 $X$. ...
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### 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 $t: S \rightarrow[0,1]$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ... 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 ...
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### 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. ...
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### 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\...
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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 $||a − b|| \leq \sup_i ||a − p_i||$....
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 ...