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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If $G$ is shellable, then $G \backslash \{x_i\}$ is shellable?

A simplicial complex $\Delta$ on the vertex set $\{x_1,\dots,x_n\}$ is shellable if the facets of $\Delta$ can be ordered, say $F_ 1 , . . . , F _s$, such that for all $1 \leq i < j \leq s$, there exists some $x \in F_…
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An exercise on components of $\mathbb{S}^2$ as a closed combinatorial surface.

Suppose that the sphere $ \mathbb{S}^2 $ is given the structure of a closed combinatorial surface. Let $C$ be a subcomplex that is a simplicial circle. Suppose that $ \mathbb{S}^2\backslash C$ has two components. Indeed, suppose that this…
Krishan Bhalla
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Why is this not a triangulation of the torus?

I refer to example 4, fig.3.6, p.17 of Munkres' Algebraic Topology. He says the given triangulation scheme "does more than paste opposite edges together". Not clear to me. For those who don't have the book to hand, a rectangle is divided into 6…
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Common subdivision of two simplicial subcomplexes (on the way to topological invariance of simplicial homology)

In Munkres' Elements of Algebraic Topology Chapter 18, he aims to show that given a continuous map $h:|K|\to |L|$, there is a well-defined map $h_*:H_p(K)\to H_p(L)$, given by $f_*\circ(g_*)^{-1}$, where $f:K'\to L$ is a simplicial approximation of…
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Action of $G$ on a Simplicial Complex vs. Action of $G$ on the Homeomorphic Space

Section 5 of chapter 6 of M.A.Armstrong's "Basic Topology" is about triangulating orbit spaces. It says that if $K$ is a simplicial complex homeomorphic to space $X$ and $G$ acts on $X$ as a group of homeomorphisms, then we can find a simplicial…
Masoud
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Elementary results on Stanley-Reisner rings

Context: I am looking for topics for an final exam talk of a commutative algebra course. I have come across the notion of Stanley-Reisner rings in the Miller and Sturmfels' book Combinatorial Commutative Algebra and Stanley's book Combinatorics and…
qualcuno
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"Simple to state, but difficult to solve" problems which require analyzing topology of simplicial complexes?

In a User's Guide to Discrete Morse Theory, Robin Forman writes: A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. In order to appreciate this statement, can I…
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Necessary condition on homology group of a set to be contractible

We call a topological space is contractible iff it is homotopic to a point. Since homology group is homotopy invariant, we can see that under any abelian group as coefficients set, a topological space $(X, \tau)$ has $H_1(X) = 0$ if $X$ is…
Sanae Kochiya
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Generalising dual triangulation of manifolds

We know the following “geometric” version of Poincaré duality: Let $M$ be a closed $m$-dimensional manifold and let $\mathfrak{X}_*$ be a finite simplicial complex with $|\mathfrak{X}_*|=M$. We can build a dual cell complex $\mathfrak{X}^*$: For…
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Kirchhoffs laws as described by homology

I was wondering what the relation between Kirchhoffs laws and simplicial homology is. The voltage law states that $\sum V = 0$ around a loop, and the current law that $\sum I = 0$ around a vertex, so it seems the voltage law is described as a…
Frank Vel
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Continuity of function between convex and compact sets

Given that $$\Delta_n=(x_1,...,x_{n+1})\in \{\mathbb R^{n+1}: \Sigma x_i=1, x_i\geq0\},$$ set $P=\Delta_m\times\Delta_n$ and $s=m+n$. Since each point of $P$ and $\Delta_s$ can be describe using $s$ parameters, they can be embedded in $\mathbb R^s$…
rgm
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Converting a $1$-connected simplicial complex into a $1$-reduced simplicial set

There is a natural way to convert a simplicial complex $C$ into an "equivalent" simplicial set $S$: after ordering the vertices, the simplices in $C$ correspond exactly to the non-degenerate simplices of $S$. If $C$ is finite and connected, it can…
Peter Franek
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Generalized ordering on simplicial complex

The vertices of simplicial complexes are usually totally ordered so that face maps of each simplex can be defined easily for the purposes of homology. That gives an "oriented" simplicial complex. But that's not always necessary, since a partial…
Herng Yi
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Showing that this particular subspace of $\mathbb{R}^2$ is non-triangulable.

Problem: Let $S=\{(x,y):0\le y\le 1, x=0\;\, \text{or}\;\, x=1/n \quad\text{for}\; n=1,2,\cdots\}\cup ([0,1]\times {0}$), Show that $S$ is not triangulable. Note: my definition for triangulation is that a set $X$ is triangulabe if there exists…
user160738
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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: Definition 1: Let $\mathcal{V}$ be a finite…
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