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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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Betti numbers of the Stanley-Reisner ring of a simplicial complex which is the cycle on $n$-vertices

Let $\Delta$ be a simplicial complex which is the cycle on $n$-vertices $V=\{x_1,...,x_n\}$ (say) i.e. the facets of $\Delta$ are $\{x_i, x_{i+1}\}$ for $1\le i\le n$ with $x_{n+1}=x_1$. Let $S=k[x_1,....
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How to prove the following homology group isomorphism?

I am new in this part. Can anyone help me to solve this question? Thanks! Let $K=K_1\cup K_2$. $K_1 \cap K_2$ is $r$-dimension. Here $K, K_1 $ and $K_2$ are all simplicial complex. Then the ...
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Elementary proposition on triangulations

I have a question on triangulations. Let T be a triangulation of a d-dimensional cross-polytope. Let s be a (d-1)simplex that does not lie on the boundary of the cross-polytope. How can we show that ...
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Show that all triangulations of a compact surface are equivalent

I'm having trouble with solving the following question: Let $T_1, T_2$ be two finite triangulations of a compact surface. Show that if $E_{T_1}\cap E_{T_2}$ is a finite set of points, where $E_{T_i}$ ...
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Proving a weaker version of the Jordan Curve Theorem (by induction)

I'm trying to solve the following question: 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 ...
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28 views

Components of $\mathbb{S}^2$ as a closed combinatorial surface.

I'm trying to solve the following problem, and I'm not having much luck: Suppose that the sphere $ \mathbb{S}^2 $ is given the structure of a closed combinatorial surface. Let $C$ be a ...
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34 views

Let $K$ be a simplicial complex (that need not be finite). Prove $|K|$ is Hausdorff.

As the title makes clear, I'm trying to solve a question which asks me to show the topological realisation of a simplicial complex is Hausdorff. The question reminds me that a subset of $|K|$ is open ...
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32 views

Poincare dual simplicial structure of complexes homotopy equivalent to manifolds

Given a closed $n$-manifold $M$, Poincare duality equips us with an isomorphism: $$H_k(M)\cong H^{n-k}(M)$$ Here I'm speaking of singular homology with coeffecient in $\mathbb{Z}_2$. Suppose now $M$ ...
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Necessary condition on homology group for 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 ...
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recovering the information of the vertices from a simplex

It seems quite obvious that, when given a simplex, its set of vertices is uniquely determined by the simplex. The formal formulation of this intuition is as follows: Suppose that the points $\{v_0,...
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Horn of a Simplicial Set

My question refers to description of horns $\Lambda^n_k$ for Kan Fibrations in Laures' and Szymik's "Grundkurs Topologie" (page 227). Sorry, there exist only a German version. Here the relevant ...
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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 $\eta_2$ ...
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Distance in simplicial complex

There is an obvious metric on a graph between vertices, and for a simplicial complex $K$ we can define a metric between two $d$ faces, by defining a graph whose vertices are it's $d$-faces and it's ...
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Problem on simplicial complexes.

If $(S_0,P_0)$ and $(S_1,P_1)$ are abstract simplicial complexes, a simplicial map from $(S_0,P_0)$ to $(S_1,P_1)$ is a function $f\colon S_0 \longrightarrow S_1$ such that, if $U\in P_0$, then $f(...
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How is the homomorphism $b: LC_n(Y) \to LC_{n+1}(Y)$ where $b[w_0\cdots w_n] \mapsto [bw_0\cdots w_n]$ well-defined?

This is on page $121$ of Hatcher's Algebraic Topology. $Y$ is a convex subset in some Euclidean space. The linear maps $\Delta^n \to Y$ generate the subgroup of linear $n$-chains on $Y$, $LC_n(Y) \...
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Is this a non-example of a $\Delta$-complex?

I know this is not a simplicial complex, but is it a $\Delta$-complex? Here is the definition of $\Delta$-complex from Hatcher: I don't believe any of the rules are broken.
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Free faces in $M_\kappa$ polyhedral complexes.

I am reading Bridson and Haefliger's ``Metric Spaces of Non-Positive Curvature" and I am struggling with Definition 5.9 of a free face. Let $K$ be an $M_\kappa$-polyhedral complex. A closed $n$-...
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Necessary and sufficient condition for simplicial complex to be subspace

Let $K$ be a simplicial complex and $|K|$ it's realization with the topology that $F\subset |K|$ is closed iff $f\cap \sigma$ is closed in $\sigma$, for every $\sigma\in K$. Then this topology ...
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How to systematically collapse a $n$-simplex to a point via elementary collapses

An elementary collapse (https://en.wikipedia.org/wiki/Collapse_(topology)) is the removal of a pair of simplices $\sigma,\tau$, such that $\dim \tau=\dim\sigma-1$ and $\tau$ is a free face of $\sigma$....
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Finite Simplicial Complexes: Show that open sets have open connected subsets containing any point

The question: I am required to show the following: Let $K$ be a finite simplicial complex, let $x \in |K|$ and suppose that $U$ is an open set containing $x$. Then there is an open connected set $V$ ...
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Simplicial and De Rham Homology on Manifolds

I'm looking for a recomendable reference/source for a rigorous proof that for manifolds (with "nice enough" structure) the simlicial and De Rham (co)homologies coincide. Especially, I know that ...
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Do subspaces of polyhedron give subcomplexes?

Let's say I have a topological space $X$ that's a polyhedron, then there exists a simplicial complex $K$ and a homeomorphism $h : |K| \to X$. If we have a subspace $A \subseteq X$, is there a ...
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Why is induced map on zero homology the identity and not negative the identity?

Suppose we have a simplicial map $f$ on a path connected simplicial complex $X$. The answer here: Induced map on zeroth homology is zero claims that the induced map on the $0$-homology given by $f_*: ...
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Existence of a type of simplicial complex

I want to prove that the following proposition is false: There exists a homologically trivial finite 2 dimensional simplicial complex $\mathcal K$ such that every edge (1 dimensional simplex) has at ...
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Relation between the number of facets and of free faces

First, give the definition. A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face) A simplex $\tau$ is called a free face if it is the face of only one ...
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Barycentric subdivision preserves geometric realization

I have the following definitions: Definition 1: A simplicial complex $K$ is a family of finite nonempty subsets of a set $V_k$ (the elements of $V_k$ are called vertices) such that: 1) if $v\in V_k$,...
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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 ...
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74 views

Homology groups with different complexes

When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes. Does it matter which one I use in general? Do they always yield the same homology ...
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1answer
39 views

A non-polyhedral pair $(X,A)$ with both $X$ and $A$ polyhedral

Does there exist a topological space $X$ and a closed subspace $A$ such that each of $X$ and $A$ is isomorphic to the topological space of some simplicial complex, but such that there does not exist a ...
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1answer
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Property of a point in a finite simplicial 3-complex

I've got a homework problem this week about proving that any finite simplicial 3-complex with the property that each point in the complex is contained in an open set homeomorphic to an open set $U$ in ...
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2answers
55 views

Homology of a simplicial complex union a simplex

Let $L=K\cup\sigma$, where $L, K$ are abstract simplicial complexes, and $\sigma$ is a simplex whose faces are all in $K$. (This condition is necessary for $L$ to be a simplicial complex.) ...
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1answer
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why the boundary of loop is zero?

In simplices complex, it is known that boundary of boundary is zero $$\partial_{n-1} \partial_{n} \sigma_{0,\ldots,n} = 0$$ where $\sigma_{0,\ldots,n}$ is a n-simplice consists of $n+1$ point, which ...
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Alexander duality on (co)chain level

In the book by Stöcker & Zieschang, the Poincaré duality is obtained by an isomorphism $\gamma \colon \operatorname{Hom}(C_q(K), \mathbb{Z}) \to C_{n-q}^{\ast}(K^{\prime})$, $\gamma(\varphi) = \...
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Lower bound for intersection of a ball near the boundary of a bigger ball

Let $\mathbf{x}=(x_0,\ldots,x_k )\in (\mathbb{R}^d)^{k+1}$ where $1\leq k\leq d$ such that the points $\{x_0,\ldots,x_k\}$ are in general position. Thus $\mathbf{x}$ defines a unique $(k-1)$-sphere ...
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58 views

Is this a triangulation for the 2-torus?

I am not quite sure I understand simplicial comlexes/triangulations. For instance, I think that the below image represents a triangulation for the 2-torus. Am I correct?
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n-skeleton product preserving.

It is true that the n-skeleta functor $$sk_{n}:sSet\rightarrow sSet $$ is product preserving i.e., $sk_{n}(X\times Y)$ is naturally isomorphic to $sk_{n}(X)\times sk_{n}(Y)$.
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1answer
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Two-fold branched cover of Hexagon - Which map is meant

I am trying to understand an obvious example in a paper but do not get what is meant by: "X is a hexagon and $f:X \rightarrow \sigma^2$ is a two-fold branched cover (branched at the center of $\sigma^...
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1answer
76 views

Simplicial approximation to CW Complex

In Hatcher's book, the following theorem is mentioned (Th 2C.5, page 182): Every CW complex $X$ is homotopy equivalent to a simplicial complex, which can be chosen to be of the same dimension as $X$...
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Is the simplicial complex with vertices the critical points of a $C^2$-function on a Hilbert manifold $X$, homotopy equivalent to $X$?

Let $X\subset\mathbb{R}^{\infty}$ be a paracompact connected Hilbert manifold. Moreover, let $f:X\to\mathbb{R}$ be a $C^2$-function. Then is the simplicial complex $\Delta:=\{p\in X:df(p)=0\}$, ...
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How many cells do we need in $\mathbb{S}^n$ to induce $\pi_n(\mathbb{S}^m)$?

Take on the spheres $\mathbb{S}^n$ and $\mathbb{S}^m$ some "easy" simplicial structures $\Sigma^n$ and $\Sigma^m$ (e.g. as surfaces of the corresponding simplices). Then by simplicial approximation ...
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1answer
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Question about Hatcher's proof of equivalence of simplicial and singular homology for $\Delta$-complexes.

In Hatcher's Algebraic Topology, the following argument appears (note that this is only part of the theorem). I see why the diagram commutes (with respect to to homomorphism induced by the "inclusion" ...
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Proof of the general case of Feynman's integration trick

I want to show that $$\frac{1}{\prod\limits_{i=0}^{i=n}A_{i}}=n!\int\limits_{\mid \Delta^{n}\mid}\frac{d\sigma}{\left( \sum s_i A_i \right)^n}$$ Where $d\sigma$ is the lebesgue measure on the ...
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1answer
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Product of Semi Simplicial Sets

Is it true that given semi-simplicial sets $X,Y$, then $|X|×|Y|$ has a natural semi-simplicial structure? I would guess that it is true, but I cannot find anything about it online. I know that the ...
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39 views

is the square of the boundary matrix zero?

let $K= \{\sigma_1...\sigma_n\}$ be a simplicial complex with n simplices with faces of the ith simplex having lower index than i. Let the field of coefficients be $\mathbb{Z}_2$ recall the boundary ...
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Simplicial intersection product

Let $X$ be both a simplicial set and a closed $n$-dimensional manifold. We have a duality isomorphism $H^k(X)\to H_{n-k}(X)$. Furthermore, let $Y,Y'\subseteq X$ be two subcomplexes and closed ...
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Covering space of simplicial complexes

Is there a reference that discusses covering spaces of (abstract) simplicial complexes? Is there an alternate definition of covering spaces for (abstract) simplicial complexes? The traditional ...
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1answer
85 views

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 ...
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How can we interchange the colors of districts in 4 color problem

In the paper On the geographical problem of the Four Colors by Kempe, at the end of page 194, it is stated that It will readily be seen that we can interchange the colours of the districts in one ...
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1answer
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compute f-and h-vector of simplicial complex

A simplicial complex $\Delta$ is uniquely determined by its facets $\mathcal{F}(\Delta)$. I know how to compute the $f$-vector and therefore the $h$-vector given all the faces of $\Delta$. Now given ...
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How do I calculate the boundary of an $n$-simplex?

On a paper I am reading, it states that the boundary of an $n$-simplex $\Delta$ with points $P_0, P_1, \ldots, P_n$ is given by the formal sum of its oriented $(n-1)$-dimensional 'faces' $$\partial\...