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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Universal property for chain maps

Given simplicial complexes $X,Y,Z$, such that there exists a chain map $f: C_*(X\times Y)\rightarrow C_*(Z)$, and $\pi_Z$ is the projection $C_*(Z\times Y)\rightarrow C_*(Z)$, does there exist a unique chain map $f':C_*(X\times Y)\rightarrow…
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Topology of simplicial complexes

I'm trying to prove that the topology of $|K|$ as a subspace of $\mathbb{R}^{m}$ is the same as the topology of $|K|$ obtained as a quotient space. I was thinking on the equivalence relation of that identification in this way: $ \sigma$ ~ $\tau \iff…
user381400
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Hatcher exercise: 2.1.1

I have already seen an answer to this question, which both places I have found it, it was done pictorially, something I consider useless due to a lack of scalability.(How to learn to deal with problems in the scaling where such pictures fail if you…
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Example of homology group of a simplicial complex.

I need a little help with understanding homology groups. In particular, consider a simplicial complex with three 1-cycles and one 1-boundary (so we have two 1-holes?). Then the first homology group is the second order cyclic group, i.e.…
Kosm
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Morphisms induced between chain groups commute with boundary operators: proof?

Definition An abstract simplicial complex $K$ is a pair $(X_K,\Phi_K)$ where $X_K$ is a set of points, and $\Phi_K\subseteq2^X$ such that: If $S\in\Phi_K$, then $2^S\subseteq\Phi_K$; $\{x\}\in\Phi_K$ for all $x\in X_K$. An ordered abstract…
MickG
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Demonstrations on the Simplicial complex of Graph

where I cannot understand $F\in\Gamma\land G\subseteq F\Rightarrow G\in\Gamma$. I would like to see an example about the simplicial complex of a graph such as a cycle graph $C_3$. What are demonstrations such as $\Gamma (C_3)$ about $\Gamma$?
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Is the simplicial join of two spherical simplicial complexes itself spherical?

I think this ought to be true, but I am struggling to see why. Of course if one of the spheres is $S^0$ then this is trivially true, as we are just glueing two cones along their boundary. I'm not seeing a general argument though.
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Fundamental Groups of a Simplicial Complex and the Underlying Space

Now I know how the fundamental group of a Simplicial Complex is defined, as well as that of a Toplogical Space. Could someone explain the process of how we prove that for a simplicial complex $X$ , how $\pi_1(X)$ and $\pi_1(|X|)$ are isomorphic?
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Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a closed surface $S$. If $P$ has $f $ faces, $ e$…
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Example that the topology of $|K|$ is larger than the topology $|K|$ inherits as a subspace of $\mathbb{R}^n$

I'm a lil bit confused with the example 3 from Munkres, that the topology of $|K|$ is larger than the topology $|K|$ inherits as a subspace of $\mathbb{R}^n$. Let $K$ be the collection of 1-simplices $\sigma_1,\sigma_2,...$ and their vertices, where…
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Definition of a "n-"simplex

I have a simple question about the definition of a simplex. In this paper by Jonathan Huang he notes that every point on the simplex can be expresed as a linear combination $$x=\sum\limits_{i=0}^n \alpha_i p_i$$ with $\sum\limits_{i=0}^n \alpha_i…
user203327
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Simplicial Complexes and their Skeletons in Homology

Let $K$ be a simplicial complex and $K^r$ its $r$-skeleton, I wanto to show that the simplicial homology groups $H_p(K)$ and $H_p(K^r)$ are isomorphic for all $0 \leq p \leq r-1$. And also I wonder if there is a relation between $H_r(K)$ and…
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Extend simplicial homeomorphism in a PL surface

Let $S$ be a connected PL closed surface. How can I show that, given a 2-simplex $\Delta$ in $S$ and a simplicial homeomorphism $g:\Delta\to \Delta$ that preserves orientation, this can be extended to a simplicial homeomorphism $h:S\to S$ ? EDIT:
user72870
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Why are simplicial complexes modules?

When a simplicial complex is viewed as a R-module (such as appears in an exact sequence), what is the corresponding ring R and the Abelian group M?
Yan King Yin
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A simplicial complex with vanishing first homology but nonzero fundamental group

I'm interested in the simplicial complex and I do not know much of algebraic topology to use the answers that…
Vasco
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