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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PL-collapsibility of simplicial complexes

I am currently working on the PL-collapsibility of a particular finite simplicial complex. Here is my definition of "PL collapsible": I say that there is an elementary collapse from $X$ to $...
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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 ...
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Confusion about the behavior of $p$-simplices under “inversions” (reversing ordering of indices)

I have found a very nice paper which generalizes much of vector calculus to discrete lattices through the use of simplicial complexes, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.59.1217 ...
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Krull dimension and depth of face rings of small complexes

I'm trying to calculate Krull dimension and depth of face rings for small simplicial complexes (up to 10 vertices). Here are 2 examples of such rings: a) $K[x, y, z, t] / (xy, z)$ b) $K[x, y, z, t] / (...
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weak equivalence of the geometric realisation of a total singular complex and a topological space (from P.May concise course in Algebraic Topology)

In P.May's book "A concise course in Algebraic Topology", chapter 16, He establishes a weak equivalence between $\Gamma X = |S_*(X)|$ and $X$, where $X$ is a topological space, $S_*(X)$ is ...
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Reciproque image of simplex [closed]

Let $M=B\times S^{1}$ be the solid torus where $\partial M=X\times F= S^{1}\times S^{1}$. We consider the projection $\pi : \partial M \longrightarrow X$ which induces the simplicial map $$\pi_{*} : ...
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Snake lemma, pictorially on simplicial complexes

I'm trying to understand what the snake lemma 'computes' on small examples. Consider this: It seems to me that given two 'defective' / 'incomplete' simplicial complexes that are related through a ...
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Orientability of simplicial complex

Suppose $K$ is an abstract simplicial complex. We know $K$ has a geometric realization |K|, Suppose $|K|$ is a manifold, Is there any necessary and sufficient condition on $K$ for orientability of |K|?...
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How can I construct these homeomorphisms?

From Rotman's Algebraic Topology: If $X$ is a polyhedron and $x \in X$, there exists a triangulation $(K,h)$ of $X$ with $x = h(v)$ for some vertex $v$ of $K$. I'm having difficulty figuring out how ...
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Trying to track down Sperner's lemma with signed counting of triangles

40 years or so ago, a kid named Jeremy showed me a proof of the two-dimensional Brouwer fixed-point theorem, which used what I have since come to know is called "Sperner's lemma." The two-...
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Simplicial approximation iff $x \in |K|$ and $f(x) \in s^{\circ} \Rightarrow |\psi|(x) \in s$

From Rotman's Algebraic Topology: Prove that a simplicial map $\psi : K \rightarrow L $ is a simplicial approximation to $f : |K| \rightarrow |L|$ if and only if, whenever $x \in |K|$ and $f(x) \in ...
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Simplicial Stokes theorem

I have been intrested lately in a simplicial version of Stokes theorem, which intuitively I think should be true in simplicial settings. I think if $X$ is a simplicial complex and $X(k)$ are its $k$-...
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Proving simplicial homology is preserved on mapping?

Assume we have a the $\mathbb Z\text{ modules}$ $S1 \equiv (F, E, V)$ with boundary maps $(\partial_{FE}: F \rightarrow E$, $\partial_{EV}: E \rightarrow V)$, with the condition that $\partial_{EV} \...
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Why is the Sullivan-Cohen-Sato class well defined?

Let $K$ be a triangulation of an $n$-manifold $M$ (for $n \geq 5$). (That is, $K$ is a simplicial complex whose underlying topological space is $M$.) The Sullivan-Cohen-Sato class of $K$ is defined to ...
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How to divide a unit space into many simplices?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
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80 views

Prove that the star of a point is an open subset of underlying space of a simplicial complex

From Rotman's Algebraic Topology Prove that $\text{st} (p)$ is an open subset of $|K|$, where $|K|$ is the underlying space of a simplicial complex and $\text{st} (p)$ is defined as $\bigcup_{s \in ...
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How is a simplicial complex completely determined by a simplicial set?

I was wondering how a simplicial complex is completely detremined by a simplicial set. I say let $K$ be a simplicial complex. Then $K=(V,S)$ where $V$ is a set and $S$ is a collection of finite non-...
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What does the basis of $H^{\Delta}_1(T)$ look like? where $T$ is the Torus.

Using a $\Delta$-complex structure, I've calculated the simplicial homology groups of a Torus. I used 2 2-simplices, 3 1-simplices and 1 0-simplex under the quotient map. I obtained the following ...
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Example of non-contractible acyclic finite simplical complex

From the article "f-vectors of acyclic complexes" (Discrete Math. 1985) by Gil Kalai, we know that the relation between f-vectors of an acyclic simplicial complex. In other words, one can construct ...
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$\Delta$-complex structure of $S^1$

I've just started reading about algebraic topology. I've read the definitions of a $\Delta$-complex structure. I have a couple questions: How would one construct one for $S^1$? I can see that if we ...
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On the geometric realization of a finite abstract simplicial complex which is connected, orientable $3$-manifold without boundary

Let $\Delta$ be an abstract simplicial complex on finitely many vertices and $|\Delta|$ be it's geometric realization. (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex) If $|\Delta|$ is ...
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Tutte–Grothendieck invariants and Tutte polynomial. Is this property even used in the proof?

I am using “Matroids: A Geometric Introduction” by Gordon and McNulty. On page 337, in Theorem 9.19 Tutte-Grothendieck invariants are defined. The theorem itself gives their description in terms of ...
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Dyadic division of a simplex - is it possible in dimension > 2?

Assume T is a triangle. We can join the middle points of the sides of T, and this form a division of T into 4 triangles, homothetic to the original triangle, and with sides equal to sides(T)/2. Let me ...
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Degree of vertices in a barycentric subdivision

Consider the barycentric subdivision of a triangulation of a manifold $X$, possibly with boundary. I am interested in the degree of vertices after subdivision mod $2$. I think it is true that for ...
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A lemma used for proving the “The simplicial approximation theorem”

I'm reading on Homology groups from Munkres' "Elements of algebraic topology".While proving lemma 16.3 , on pg 92 and 93 , Munkres defines a map $\rho :{\mid B\mid}\times I$ $\rightarrow$ $\mid K\mid$ ...
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Simplicial complexes embedded on a compact manifold

Every finite graph can be embedded on some compact surface of sufficient genus such that no two edges cross. If $S$ is a finite simplicial complex of dimension $n$, can $S$ be embedded in some ...
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Computing the homology of two simplicial subcomplexes of $K$

Let $K$ be a simplicial complex on a vertex set $V=\{u_1,\dots,u_n,v_1,\dots,v_m\}$with facets given by the sets $$\sigma_i=V-\{u_1,u_i\},\tau_j=V-\{v_1,v_j\}$$ for each $i,j$. I want to prove that ...
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Approximating curvature of a Riemann Manifold

I have read that given a 2-manifold $M$, it is possible to approximate the Gaussian Curvature at any point $p$ in $M$ using simplicial complexes made of triangles in such a way that as the maximum ...
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Square-free monomial ideal, Cohen-Macaulay ring and ideal quotient

Let $k$ be a field and consider a polynomial ring $R=k[x_1,...,x_d]$. Let $I$ be a square-free monomial ideal with $\mu(I)=t$ and let $I=(f_1,...,f_t)$ , where $f_1,...,f_t$ are square-free monomials....
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Subcomplexes of a Closed Combinatorial Surface

I know that a requirement for a finite connected simplicial complex $K$ to be a closed combinatorial surface is that the link of any vertex is a simplicial circle. So, suppose that $C$ is a subcomplex ...
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if $H_1(X)$ has torsion in finite simplicial complex complex $X$ then a particular embedding into $R^3$ does not exist

I was going through exercises in Hatcher to prepare for quals and then came across the following: (Ex 2.2.35) Use Mayer-Vietoris to show that a finite simplicial complex $X$ for which $H_1(X)$ ...
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Persistent homology has to be free, right?

I've been convinced that the homology groups you get when computing the persistent homology of a data cloud have to be free. But now I'm second guessing myself. Can we quickly say why this has to be? ...
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Examples of generalized homology spheres which are not triangulations of spheres

A simplicial complex $\mathcal{K}$ is a generalized homology $n$-sphere if the following hold: $\mathcal{K}$ has the same homology as $S^n$ For each non-empty simplex $\sigma \in \mathcal{K}$, $\...
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Simplicial map of the circle $S^1$

I want to construct a simplicial map $S^1 \to S^1$ of degree $n>0$ by giving a simplicial structure on $S^1$ with minimal number of vertices. (Here, the simplicial structure for the domain $S^1$ ...
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Every simplicial map is cellular.

A simplicial map is by definition a map $f:K\to L$ between simplicial complexes that sends each simplex of $K$ to a simplex of $L$ by a linear map taking vertices to vertices. A cellular map is by ...
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If $g$ is a simplicial map of a finite simplicial complex $X$, then the diagonal of the matrix of $g_*:H_n(X^n,X^{n-1})\to (X^n,X^{n-1})$ is zero

Let $X$ be a finite simplical complex, and let $g:X\to X$ be a continuous simplicial map such that $g(\sigma)\cap \sigma=\emptyset$ for every simplex $\sigma $ in $X$. Then why is the diagonal of the ...
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Can an $n$-orthoplex be shrunk to a point like an $n$-cube?

An inverse process of expanding a point to a 4-cube is shown in [https://en.wikipedia.org/wiki/Hypercube]. I assume that it can't be done for $n$-orthoplex, as having $$2^{k+1}\binom{n}{k+1}$$ $...
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Problems with simplicial space

So I'm going through "Elementary Topology Problem Textbook" by Viro and all, and have problems with 23.3x paragraph which is devoted to simplicial schemes. More concretely I can't tackle 23.4x problem ...
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Conditions for creating a void homeomorphic to $n$-cube or to $n$-orthoplex within a discrete manifold

Assume an $n$-dimensional discrete manifold $M$ ($n$-dimensional simplicial complex in which each ($n-1$)-simplex has exactly two adjacent $n$-simplices or only one if it is on a boundary of $M$) ...
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On a particular kind of simplicial complex with maximum facet size of $3$.

Let $\Delta$ be an abstract simplicial complex on $n$ vertices such that $\max \{|F| : F$ is a face of $\Delta \}=3$. Let $f_2$ be the number of faces of size (cardinality) $3$ and $f_1$ be the ...
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Understanding affine combination of points in affine plane

I have been going through Duke's series of lecture notes on computational topology, and on their formulation of simplicial complexes (https://www2.cs.duke.edu/courses/fall06/cps296.1/Lectures/sec-III-...
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Delta complex structure of $\mathbb R P^n$

Here is a $\Delta$-complex structure of $\mathbb R\text P^2$ found on the internet: I find it very difficult to see why this is homeomorphic to the sphere $S^2$ with antipodal pairs identified. I ...
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Is the induced map $BX_U\to BR_U $ a homotopy equivalence?

I'm reading Segal's paper Classifying Spaces and Spectral Sequences and in section $4$ he tries to prove for good coverings(which admits a partition of unity) $\{U_{\alpha }\} $ over a topological ...
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1answer
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Question about faces of a simplex and relation to complexes

I'm reading Hatcher's book on algebraic topology, p103: Let $[v_0, \dots, v_n]$ be an $n$-simplex. A face of $[v_0, \dots, v_n]$ is the $(n-1)$-simplex obtained by deleting one vertex $v_i$ from the ...
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Generalization of the independence complex of a graph to simplicial complexes

Let $G$ be a graph, then the independence complex of $G$, which we denote $I(G)$, is the set of subsets of the vertex set of $G$ such that no two vertices within a set in $I(G)$ are connected by an ...
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Proof that the edge group $E(K,v)$ of a simplicial complex is isomorphic to the fundamental group $\pi_1 (|K|,v)$ of its polyhedron.

I'm reading this proof of the statement above, and get stuck on the part about the surjectivity of the homomorphism $\phi$. My understanding of the argument is that, given a loop $\alpha$ in $|K|$, ...
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1answer
55 views

Homology groups of a square

Consider the unit square with a $\Delta$- complex structure obtained by the two $2$-simplices $[v_0,v_1,v_2], [v_2,v_0,v_3]$. Is $\Delta{}_0(X)\cong\mathbb{Z}^6$ or $\mathbb{Z}^4$ where $X$ is the ...
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Is this a valid definition of $\Delta$ complex

I am trying to understand what delta complexes are: Is the below definition correct? $X\text{ is a }\Delta\text{-complex if }X=\bigcup_{i=0}^\infty X^i\text{ where }X^i=\bigcup_{\alpha\in A_i}\Delta_\...
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1answer
28 views

How many abstract simplicial complexes are there on an $n$-set?

I was wondering how to count the number of abstract simplicial complexes there are on $\left[n\right]=\left\{1,...,n\right\}$. An abstract simplicial complex being a set $\Sigma\subset\wp\left(\left[n\...
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1answer
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On the proof of the simplicial decomposition theorem

My question is on the proof of the simplecial decomposition theorem as found in the book by Fuchs and Fomenko in page 29. Specifically, I am struggling with the proof that the mapping the authors ...

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