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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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 ...
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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 $||a − b|| \leq \sup_i ||a − p_i||$....
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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How is a quotient simplicial complex by a group action defined?

Suppose we have a simplicial complex $K=(V, S)$ a simplicial left action of a group $G$ on $K$, i.e. an action of $G$ on the set $V$ of vertices of $K$ with the property that $\sigma \in S \implies g \...
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Why is simplicial approximation a powerful definition?

When extending homology from chain complexes to the polyhedron of a simplicial complex, we use the simplicial approximations to prove invariance of homology. This eventually leads us to show that ...
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Dimension of the union of a 4-dimensional and 3-dimensional simplex

Say I'm given two simplices: one is a 4-simplex and the other is a 3-simplex, and I've proven their intersection is non-empty. How do I now explain what the dimension of their union is? I understand ...
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Are loops not allowed when gluing together simplices in a simplicial complex?

I'm taking a graduate geometric topology class and our professor made a quick remark that we're not allowed to make stuff like these by gluing together simplexes in a simplicial complex. I didn't get ...
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When can 2 different simplicial chains share the same boundary?

Suppose I have a filtration of a simplicial complex $K$: $$ \emptyset=K_0\subseteq K_1\subseteq K_2\subseteq,...,\subseteq K_n=K $$ Suppose $\sigma_j$ is a $d$-dimensional simplex that first appears ...
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Counting birth and death classes of persistent homology.

Suppose we have a filtration of simplicial complexes $\emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K$. Then, for $i \leq j$, we have induced homomorphisms $f^{i,j}_p \colon H_p(K_i) \...
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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 ...
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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 $\{v_0,\dots, v_n\}\in K$ ...
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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 $\alpha\leq\beta$...
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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) ...
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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:...
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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 $f$-...
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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 ...
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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 ...
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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 ...
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Geometric Realization of Standard Combinatorial $n$-Simplex is Standard Topological $n$-Simplex

Let $\Delta$ be the simplex category with objects $[n]=\{0,...,n\}$, $n\geq 0$, and morphisms ordering preserving functions $[n]\rightarrow [m]$. Then let the standard categorical $n$-simplex be $Hom_{...
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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 ...
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What is a flat Kan fibration?

What is the Kan fibration analog of a connection? of being `flat'? A Kan fibration is the simplicial analog of a topological fibration. What structure on a Kan fibration is the simplicial analog of ...
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Why if $n$ is odd the simplicial homology is $\mathbb Z,$ while if $n$ is even it is $0$?

I am trying to solve this question: Compute the homology groups of the $\Delta$-complex $X$ obtained from $\Delta^n$ by identifying all faces of the same dimension. Thus $X$ has a single $k$-simplex ...
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Constructing connected complexes with prescribed homology groups

This exercise is taken from Munkres' Algebraic Topology book, it is in Section 6 of Chapter 1. The problem asks you to find a $2$-dimensional complex $K$ with underlying space $|K|$ connected and $H_1(...
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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 ...
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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 ...
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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). ...
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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, ...
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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 ...
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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 ...
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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 ...
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Proof of discrete Hodge decomposition

In this survey by Lubotzky, he has the following: Proposition 2.1 (Hodge decomposition): The following are true: $C^i=B^i\oplus\mathcal{H}^i\oplus\mathcal{B}_i$, $\mathcal{H}^i\cong H^i(X;\mathbb{R})...
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Computing Simplicial Homology Groups of a structure

Take the unit disk $D^2$ and take the equivalence relation on the boundary ($S^1$) by identifying the following arcs: $(0, 2\pi/3), (2\pi/3, 4\pi/3), (4\pi/3,2\pi)$ (alternatively identifying points $...
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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 ...
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Covering $n$-simplex with $k$-subsets to produce a lower $m$-simplex, $m<n$?

Let vertices of an $n$-simplex be labeled $\{x_1,x_2,...,x_n\}$ and let the $k$-subsets or $k$-intersections ($k \leq n$)be identified as $x_{i_1} \cap x_{i_2} \cap ... x_{i_k}=x_{i_1}x_{i_2}...x_{i_k}...
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Can $H_1(\partial \lvert \mathcal R \rvert ) \to H_1(\lvert \mathcal R \rvert)$ be non injective?

Suppose we are in the following situation: we have (the image of a) closed simple curve $C$ in the plane; we have a simplicial complex $\mathcal R$ over a finite set of vertices such that the ...
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Injective objects in the category of non-negative cochain complex

I am reading a notes of Brian Conrad on hypercovering, and in the discussion on Dold-Kan correspondence, it is claimed (I might misunderstand) that the injective objects in the category of non-...
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Is every oriented simplicial complex a $\Delta$-complex?

The definition of simplicial complexes allows decomposing the above figure by $2$-simplicies $[a,b,c],[c,d,a]$. But the definition of $\Delta$-complexes in Hatcher's algebraic topology does not allow ...
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Simplicial homology of $\mathbb{Z}^2$

Imagine the grid $\mathbb{Z}^2$ as a simplicial complex such that each square is triangulated into two triangles, that is, the simplices are given by $\{(a, b),(a + 1, b),(a + 1, b + 1)\}$ and $\{(a, ...
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Hilbert series of $\mathbb Q[x_1,...,x_n]/I_\Delta+(x_1^2,...,x_n^2)$ , for Stanley-Reisner ideal $I_\Delta$

Let $\Delta$ be an abstract simplicial complex on vertex set $\{x_1,...,x_n\}$, fix the field $\mathbb Q$ and let $I_{\Delta}$ be the Stanley-Reisner ideal of $\mathbb Q[x_1,...,x_n]$ , and $\mathbb Q[...
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What is meant by compatibility of orientation of $2$-manifold and a graph on that manifold?

Situation is the following. I have a genus $g$ surface $M_g$ and a graph $G$. There is an embedding $f: G \to M_g$ and, actually, $G$ triangulates $M_g$. Our graph has a cycle $C=(v_1,v_2,v_2)$ (graph ...
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Proof of Nielsen–Schreier Theorem using Covering Spaces

I have two questions about the proof of Theorem VI.34 in these lecture notes. I have typed all that is relevant, below. Theorem. (Nielsen-Schreier) Any subgroup of a finitely generated free group is ...
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Computing Simplicial chains (and Homologies)

I'm attempting to compute the Betti numbers for the closed orientable genus 2 surface, and I'm realizing I'm completely lost. I guess to begin, I need to compute the simplicial chain complexes, then ...
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Constructing $|K|$ given an abstract simplicial complex $K = (V,\Sigma)$

On Pg. $9$ of these notes, Lackenby introduces Abstract Simplicial Complexes. The topological realization of a simplicial complex $K$, denoted $|K|$, is given by the following recipe: The author ...
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Can there be arbitrary vertices on the edge of an abstract simplicial complex?

$$P = \{a, b, c\}$$ $$R_t(P) = \{\{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{c, a\}, \{a, b, c\}\}$$ For a Rips complex $R_t(P)$, can there be other points(vertices) on the edges({a, b}, {b, c}, {c, a})...
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