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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Closure operator over simplicial complex
A simplicial complex (sometimes refereed to as abstract complex or abstract simplicial complex) is a set system $(S, \Delta)$ where $S$ is a set and $\Delta\subseteq \mathcal P(S)$ is a family of ...
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Does the long exact sequence of a pair for a simplicial complex and a face delete split when the connecting maps are zero?
Given a simplicial complex $\Delta$ and a face $\sigma\in\Delta$, define the face delete $\Delta_0=\{\tau\in\Delta:\sigma\not\subseteq\tau\}$. Then $\Delta_0$ is a subcomplex, so we get the long exact ...
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Fast Algorithms for Generating Simplicial $n$-Spheres on $k$ Vertices
As stated in title, I wonder know if there exists any algorithms for generating simplicial $n$-spheres on $k$ vertices.
The motivation for seeking such algorithms comes from my goal to compute the ...
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What is the Alexander duality map for simplicial complexes, explicitly?
Let $T$ be a triangulation of $S^{d+1}$. There is a cell complex $\bar{T}$ homeomorphic to $S^d$, whose $k$-cells are in bijection with the $d-k$-cells of $T$, and its boundary operator $\bar{T}_k$ is ...
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Triangulation of $RP^2$
This an exercise from Munkres' Algebraic Topology. Are figures 3.9 and 3.10 both triangualtions of $RP^2$, or is 3.10 different? (If yes, then it is deceptively simple).
Is it so that only the ...
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Properly Discontinuous Action of a Group on a Tree
Exercise 10.33 in Rotman's "An Introduction to Algebraic Topology" asks us to prove that
Let $G$ be a group. If there exists a tree $T$ on which $G$ acts properly (discontinuously, in the ...
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Simplicial homology does not always count holes?
Consider a triangular region (2-cell). The boundary triangle (1-cell) is a boundary because it bounds the 2-cell in the sense that there exists a neighborhood that which belong to the 2-cell and ...
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Hauptvermutung for a ball
This question is probably known for any expert in geometric topology:
Is it true that any two triangulations of the $d$-dimensional ball have a common refinement?
As I've learned, this question is ...
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When can compute homology of CW-complex using simplicial homology method, by pretending each $e^n_\alpha$ is a simplex?
I'm wondering When can we compute homology of CW-complex using simplicial homology method instead of map degree, by pretending each $e^n_\alpha$ is a simplex? e.g. like the accepted answer by @Fly by ...
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Contiguous maps induce equal maps on the simplicial homology level [closed]
Definition (Simplicial map). A map $f: K_1 \rightarrow K_2$ is called simplicial if for every simplex $\left\{v_0, \ldots, v_k\right\} \in K_1$, we have the simplex $\left\{f\left(v_0\right), \ldots, ...
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A simplicial complex $\Delta$ is flag if all minimal non-faces of $\Delta$ have cardinality 2
I come from graph theory and have some fun with the definition in the title and some other statements being made in the context of (abstract) simplicial complexes:
I understand simple graphs and a ...
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Looking for a reference/ further examples for small, simple graphs whose clique complex is a triangulation of the 2-sphere aka flag 2-spheres
... aka flag triangulations of the 2-sphere.
Quoting wikipedia, 'The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the ...
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Name of simplicial complexes with removed open simplices
I would like to know if there is a name for the following type of topological spaces: simplicial complexes with some open faces removed.
Examples.
1. A half-open interval is a 1-dimensional simplex ...
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Span of vertices in a simplicial complex
I am a bit confused on the definition of the "span of some vertices in a simplicial complex."
I am currently reading the paper Intersection Homology by Mark Goresky and Robert Macpherson. In ...
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Wondering how to correctly 'define' a topology from a graph: Is this graph a sphere?
Is below the mathematically correct way talking about a triangulation and a topology 'defined' by a graph?
I have this simple graph ...
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