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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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 map of abstract simplicial complexes (for any…
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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, I need an auxiliary result. Let $\Delta$ denote a…
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Is there a triangulation of a closed surface with each vertex incident to $n\ge 7$ triangles?

Fix some $n\ge 7$. Is there a finite 2-dimensional simplicial complex with each vertex is incident to exactly $n$ triangles, and the complex forms a closed surface. If the surface is not closed, then uniform tilings of the hyperbolic plane do the…
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(elementary?) proof of simplicial homology groups of $\Delta_n$

Using the (trivial) CW-complex structure of $\Delta_n$, I would like to compute the homology groups of $\Delta_n$. It's obvious that (for $k \leq n$) $C_k \simeq \mathbb{Z}^{{n+1 \choose k+1}}$, and that $im \delta_{k+1} \subset ker \delta_k$. It's…
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Can I Approximate a Homeomorphism by Permuting a Grid?

Let $f: [0,1]^2 \to [0,1]^2$ be a homeomorphism of the square. By the n-grid I mean the collection of smaller squares $$[a/2^n,a/2^n + 1/2^n] \times [b/2^n,b/2^n + 1/2^n] \text{ for } a,b = 0,\ldots, 2^{n}-1$$ for some natural number $n$. I would…
Daron
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Higher cup-1 product of coboundaries is also a coboundary?

In the cohomology or the group cohomology theory, suppose $\mu_1$ and $\mu_2$ are coboundaries of arbitrary dimensions, $$ \mu_1=\delta \eta_1 $$ $$ \mu_2=\delta \eta_2 $$ where $\eta_1$ and $\eta_2$ are their lower dimensional split…
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Simplicial complex with fundamental group $\mathbb{Z}/3\mathbb{Z}$

Is there any example of a simplicial complex with fundamental group $\mathbb{Z}/3\mathbb{Z}$? (Preferably a “simple” example with as few vertices as possible.) So far I learnt about $RP^2$, with fundamental group $\mathbb{Z}/2\mathbb{Z}$. Thanks.
yoyostein
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Understanding the $\operatorname{link}$ of a simplex $\sigma$

I have a question that goes as follows: Let $K$ be an abstract simplicial complex and $\sigma \in K$. Define the link of $\sigma$ as follows: $$\operatorname{link}_\sigma(K) := \{\tau \in K | \tau \cap \sigma = \emptyset \, ,\, \tau \cup \sigma \in…
Kocmoc
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The KKM lemma implies Sperner's lemma: a direct proof?

It seems fairly well-known that the KKM lemma of Knaster-Kuratowski-Mazurkiewicz implies Sperner's lemma (definitions below). However, I only know of indirect proofs, for instance that KKM can be used to prove Brouwer's fixed point theorem, and that…
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Product of standard simplexes is homeomorphic to a standard simplex

The standard $n$-simplex is the set $$\Delta_n=(x_1,...,x_{n+1})\in \{\mathbb R^{n+1}: \Sigma x_i=1, x_i\geq0\}.$$ I'd like to prove that $\prod_{i=1}^p \Delta_{m_i}$ is homeomorphic to $\Delta_m$, with $m=\sum_{i=1}^p m_i$. I noticed that it is…
rgm
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Triviality of relative cup product $H^2\times H^2\to H^4$ for spaces embeddable to $\Bbb R^4$

Let $X$ be a triangulation of $[0,1]^4$ and $A$ is a simplicial subcomplex of $X$. I would like to show that the cup product $$H^2(X,A)\times H^2(X,A)\to H^4(X,A)$$ is trivial. It is realitvely easy to show that $H^4(A)$ is trivial, which was…
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Sheaves on a Simplicial Complex

I came across a video the other day which discussed practical applications of sheaves. However, rather than defining a sheaf on the open sets of a traditional topological space, the lecturer outlined how you would define them on a simplicial…
Tac-Tics
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Simplicial Homology Does Not Depend on the Orientation

Let $K$ be a simplicial complex and denote by $K_1$ and $K_2$ the complexes obteined from $K$ with two different orientations. I want to prove that the simplicial homology groups of $K_1$ and $K_2$ are isomorphic, that is, for each $n$ we have $$…
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What is an n-cell?

In my Topology lecture notes, 'n-cell' seems to be mentioned a lot, but it never says what exactly it means. Does it mean $n$-dimensional space?
BetaY
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simplicial approximation and infinite complexes

It is well known that if $X$ is a finite simplicial complex then for every continuous map $f:|X|\to |Y|$ there exists a simplicial map $F: X^{(n)}\to Y$ that $|F|$ is homotopic to $f$. Does anyone know an example of a map $f$ where $X$ is infinite…
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