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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Why are Simplicial Chains assumed Constant? [closed]

Just curious and wondering if I am missing something obvious: Let K be a simplicial complex. Why are Simplicial chains with coefficients in a group G described as $$ g \sigma_i ^m $$ for g in G and $ \...
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Homotopy on abstract simplicial complexes vs simplicial sets

Suppose $\Sigma$ is an abstract simplicial complex, suppose the vertices are ordered, and let $\tilde{\Sigma}$ be the corresponding simplicial set, where the set of $k$-simplices of $\tilde{\Sigma}$ ...
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What is the motivation for defining and working with the simplex category?

The simplex category $\mathbf{\Delta}$ is, for our purposes, the category whose objects are $[n]=\{0,1, \dots , n-1, n\}$ for each $n = 0,1,2 \dots$, and whose morphisms are (all) order-preserving ...
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Which simple data structure may be used to represent a simplicial complex?

Graph can be represented by 1) square incidence matrix (whose dimension is the number of vertices) or 2) list or adjacency lists. My question is - is there such simple representation (e.g. by matrix ...
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Understanding homology groups and generators of a torus.

It is mentioned here in 4x how for a torus $\mathbb{T}^2$ one has the homology groups of $H_0 \cong \mathbb{Z}$, $H_1 \cong \mathbb{Z}\oplus \mathbb{Z}$ WITH generators $\{a+b+c\}$ and $\{d+e+f\}$, as ...
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39 views

Persistent Homology Betti Numbers definition

shifting from standard simplicial homology to persistent homology, there is something that I don't understand. In simplicial homology one builds a chain complex of the form $$\dots \rightarrow C_n(K) \...
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Manifold homotopic to product of two graphs

Consider two graphs $G_1$ and $G_2$, thought of as $1$-dimensional CW complexes. Suppose that $X = G_1\times G_2$ has the homotopy type of a manifold (with boundary). What can be said about the graphs ...
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24 views

Computing Persistence Diagram in a Persistence Homology Framework

I was recently reading with interest the following paper:https://arxiv.org/pdf/2102.07835.pdf and, going to appendix to retrieve some general notions of TDA, I've been stuck for a while trying to ...
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What is the reason for this step in proving the Borsuk-Ulam Theorem (by triangulation)

I'm reading Using the Borsuk-Ulam Theorem, which presents several proofs of the theorem. Since this is a book for combinatorialists, the first involves a triangulation $\mathsf{T}$ of the $n$-...
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How is the Cech complex a subcomplex of the Vietoris Rips complex?

Wikipedia states: The Čech complex is a subcomplex of the Vietoris–Rips complex. And they reference Ghrist which essentially makes the definitions: Given a finite point cloud $X$ and $\epsilon >0$,...
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Every map of a sphere can be homotoped to a map whose fiber is finite: using simplicial approximation

Let $f:S^n\to S^n$ be an arbitrary continuous map. It is a part of Exercise 15 in Hatcher's AT, chapter 4.1, to show that $f$ is homotopic to a map $g$ such that there exists a point $q\in S^n$ with $...
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Determining $1$-th simplicial homology group of the Möbius strip.

Let $\underline {\mathcal M}$ be the underlying s$\Delta$-set of the Möbius strip. I am trying to compute the simplicial homology groups of $\underline {\mathcal M}.$ I find that the $0$-th ...
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Relative homology and triangle

I'm studying about relative homology. And I want simple example. Let me consider triangle (simplicity complex): \begin{align} K=\{p_0,p_1,p_2,(p_0p_1),(p_1p_2),(p_2p_0),(p_0p_1p_2)\} \end{align} This ...
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Homology of an edge with the endpoints identified

I was studying Mayer Vietoris Long Exact Sequence. To get my hands dirty I wanted to try some examples: let $L$ be two points and $K$ be two points connected with an edge, in euclidean space. My aim ...
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Question on geometric realization of the torus.

Definition $:$ A semi-simplicial set is a sequence of discrete space $\{X_n\}_{n \geq 0},$ where $X_n$ is a discrete space consisting of $n$-simplices for $n \geq 0$ together with the face maps $d_i : ...
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Hypercovers via Resolution of Singularity

I'm trying to understand the proof of Thm 4.16 in B. Conrad's notes on Cohomological Descent. A special case of this theorem is the following form: If $k$ is a field of characteristic zero and $S$ is ...
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Simplicial complex contractible but not collapsable

Exercise 8 pag. 90 of "Computational Topology" by H. Edelsbrunner: Collapsibility (three credits). Call a simplicial complex collapsible if there is a sequence of collapses that reduce the ...
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Connected Components of the Homotopy Pullback

Let $K$, $K'$ and $L$ be simplicial sets and be a homotopy pullback square. Question: I think that in this situation, the natural map $\pi_0 (L') \rightarrow \pi_0(K') \times_{\pi_0(K)} \pi_0(L)$ is ...
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Incremental Algorithm - How to compute betti numbers

Currently, I'm trying to learn a bit more about persistence homology and computational topology. While doing so, I came across a paper entitled "An incremental algorithm for Betti numbers of ...
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Can we preserve the fixed points of a homeomorphism $f:K\to K$ using a simplicial approximation?

Let $K$ be a finite simplicial complex and $f:K\to K$ a periodic (finite order) homeomorphism. Using the simplicial approximation theorem, is it possible to obtain a simplicial approximation $g:K\to K$...
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Help proving locally constant presheaf is constant

I am reading Raoul Bott and Loring W. Tu Differential Forms in Algebraic Topology, and in the page 146 there is the next theorem: Let $\mathfrak{U}$ be a good cover on a connected topological space $...
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On the dimension of an abstract simplicial complex built from minimal vertex covers of a finite simple graph

Let $G$ be a finite simple graph on a vertex set $\{x_1,...,x_n\}.$ Let us call a vertex cover of $G$ to be minimal if none of its proper subset is a vertex cover. Let $C_1,...,C_h$ be the collection ...
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Finding Co-Differential from Differential in Homology

I am trying to use Algebra; Linear and otherwise to find the co-differential ,i.e., the differential operator d in a Cohomology Theory starting with homology. For now, I just wanted to start with a ...
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Complexity of deciding isomorphism between abstract simplicial complexes

Exercise 1 pag. 90 of "Computational Topology" by H. Edelsbrunner: What is the computational complexity of recognizing isomorphic abstract simplicial complexes? The question isn't ...
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The size of unions of some families of sets

Consider the class $\mathcal{F}_n^m$ of families of sets consisting of $n$ sets $s_i \subset \{1,2,\dots,n\cdot m\}$ of size $m$. For a family of sets $F \in \mathcal{F}_n^m$, $F = \{s_1,\dots,s_n\}$ ...
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Realizable intersection graphs

Consider (finite) symmetric weighted graphs/matrices $\{\alpha_{ij}\}$ with $\alpha_{ij} \in \mathbb{N}_0$, $\alpha_{ij} = \alpha_{ji}$, and $\alpha_{ij} < \alpha_{ii} \neq 0$ for $i \neq j$ which ...
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Can every simplicial complex be given the structure of manifolds?

I know some manifolds can be given the structure of simplicial complex by triangulation, but what about the other way around? Can every simplicial complex be given the structure of manifolds? If so, ...
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Inner product for the vertices of a regular $q$-simplex

Let $v_1, \ldots, v_q$ be the vertices of a regular $q$-simplex, which is a subset of $\mathbb{R}^q$. Is it true that it is always possible to translate such a regular $q$-simplex in such a way that $$...
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“Dual” definition of 1-connectivity in finite simplicial complex?

I am studying the concepts of finite simplicial complexes (so please forgive for any mistakes :)) and, in particular, trying to grasp the concept of higher-order connectedness. I have the following ...
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Triangulation of the topologist's sine curve

Let $X$ be $\{(x,\sin{\frac{1}{x}})\mid x\in(0,1]\}$ together with the vertical set of accumulation points, the vertical line segment at $0$. This is closed subspace of $\mathbb{R}^2$. Is there a ...
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Degree vs Valence of a vertex in a graph

To my understanding, "degree" and "valence" mean the same thing: the number of edges incident to a vertex (including multiplicity for loops). Is there a difference in the contexts ...
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Irreducible triangulations of manifolds

Does there exist a closed Riemann manifold $M$, two distinct irreducible triangulations $S_1$ and $S_2$ of $M$, and a triangulation $T$ of $M$ such that there exists a sequence of edge contractions on ...
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Is every abstract simplicial complex the independence complex of a simple graph?

Given a simple (no loops, no multi-edges) undirected graph $G$ on $n$-vertices, one can assign an abstract simplicial complex known as the independence complex (https://en.m.wikipedia.org/wiki/...
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Why define simplicial chain groups as functions rather than free Abelian groups?

I am trying to understand why the usual definition of chain groups goes Define $n$-chains as maps from $n$-simplices to $\mathbb{Z}$ that vanish cofinitely Prove $C_n$ is free Abelian Why not start ...
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What closed subspace definition is used here?

Lemma 2.2 If $L$ is a subcomplex of $K$, then the polytope $\lvert L \rvert$ is a closed subspace of the polytope of $K$, denoted $\lvert K \rvert$. In particular, if $\sigma \in K$, then $\sigma$ is ...
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Equivalence of definitions of simplicial manifolds, and which ones imply “no branching”

I've found a couple of different definitions of simplicial manifolds with boundary: A pure abstract simplicial $n$-complex such that the (geometric realization of the) link of every simplex $\sigma$ ...
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How is the sum of simplices defined?

I have recently started to learn about simplices, simplicial complexes and simplicial homology. I understand that for some set of points $\{ p_0, \ldots, p_k \} \subset \mathbb{R}^n$ a simplex $\sigma$...
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Analog of $C^{\infty}$ multiplication for discrete “vector fields”

To make the problem simple, consider an undirected graph $G$ with vertices $V$ and edges $E$. A discrete "scalar function" (0-cochain) can be defined on the vertices, taking values $f(v)$ on ...
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Triangulation that includes given subcomplex

I was reading Prasolov and Sossinsky's book Knots, Links, Braids and Three-Manifolds and came across the following statement in the proof of Theorem 9.2: We can assume $L^3$ has a triangulation $K$ ...
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Bijection between functions on finite sets and increasing chains of subsets

For this post, I will use the Von Neumann definition of the Natural numbers: \begin{align*}0&=\{\}\\1&=0\cup\{0\}=\{0\}\\2&=1\cup\{1\}=\{0,1\}\\&\cdots\\n&=\{0,1,2,\ldots,n-1\}\end{...
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Dimensions of cycles and boundaries in a full simplex

$\newcommand\rk{\operatorname{rk}}$Let $\Delta_n$ denote the full $n$-simplex $\{n,\dotsc,0\}$. It is clear that there are $\binom{n+1}{d+1}$ many $d$-simplices, since a $d$-simplex corresponds to a ...
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Question on colimit over the category of simplices

I get slightly confused on the colimit definition: $$X \cong \lim_{\Delta^n \rightarrow X} \Delta^n $$ where the symbol $\lim$ is actually for denoting colimit. I know that colimit is defined for a ...
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Different Representations of Dunce Cap

I am having a bit of confusion concerning the Dunce Cap while studying simplicial homology, hope someone can help! Given a solid triangle with vertices $a, b, c$ I usually see the Dunce Cap defined as ...
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Simplicial Complex and Euler characteristic

I find myself confused with this problem: "For each integer $n$ find a simplicial complex with Euler's characteristic $n$". In the case of positive integers, the simplicial complex would be $...
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(Reference Request) Simplicial Complexes and Homology Book

A student has asked for references on simplicial complexes, and I remember a book but can't find its name. Pretty sure it was a yellow Springer book, but not sure what series - and I'm not even 100% ...
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Distances between two complexes when using Persistence Homology

I am using Persistence Homology to look at two different facebook networks. I can generate a distance matrix between individuals and then create the usual barcodes and persistence diagrams according ...
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Not-matchings of a Hypergraph

I'd like to characterize the not-matchings of an hypergraph $H$, or more exactly its complements. A not-matching is a set of edges containing at least $2$ not-disjoint edges. If $H$ is a graph this ...
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How many affine independent points could be found in $\mathbb R^n$?

I was reading about Simplicial Complexes definition in the book "Using the Borsuk–Ulam Theorem" and I confronted with affinely independent idea. So $$v_0,..., v_n $$ are affinely ...
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Geodesics on a simplicial complex compared to the underlying manifold

I am trying to understand the relationship between a simplicial complex and its corresponding Riemann manifold. Reading this post I understand that a simplicial complex is homeomorphic to a ...
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Computing the 0-dim cohomology group of a connected simplicial complex with coefficients in $\mathbb{Z}_{2}$

Let $L$ be a connected locally finite simplicial complex with coefficients in $\mathbb{Z}_{2}$. I want to prove that $H^{0}(L)=\mathbb{Z}_{2}$. We know that $H^{0}(L)$= $Ker\delta_{1} / Im\delta_{0}=...

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