Questions tagged [simplicial-complexes]

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Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields

Let $F$ be a finite field with many elements, say more than 7 for example, and $X$ be the corresponding projective plane $\mathbb{P}^2(F)$. For a set of points in $X$, if any three of them are ...
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combinatorial equivalence of abstract simplicial complexes

what is the most natural equivalence one defines on abstract simplicial complexes ? The definition is purely combinatorial of abstract SCs. It seems to me that the combinatorial equivalence is however ...
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What is the average degree of a d-simplex?

I am a beginner in network topology topics and while I was reading an article about simplicial complexes where the authors had used random simplicial complexes, I came across a formula using "...
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Is the smooth singular simplicial set of a smooth manifold a Kan complex?

It is classical that the singular simplicial set of a topological space is a Kan complex. This is elementary and already due to presumably Kan. Q: Is the smooth singular simplicial set of a smooth ...
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Annulus theorem for pseudomanifolds

Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\...
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3 votes
2 answers
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The exactness of the associated chain complex of a simplicial free abelian group over a finite set and the normalization theorem

Update: Now I know why my method fails. But I still wanna know how to work out the original question, that is to show the exactness of the chain complex $C_*(X)$ except for two positions $n=0,N-1$. ...
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2 votes
0 answers
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Do the polyhedral homologies of a polyhedron coincide with the polyhedral homologies of its subdivision?

Definition. A convex polytope is a compact finite intersection of hyperplanes in $\mathbb{R}^n$ Definition. The polycomplex is the following data set: a set of convex polytopes, closed under ...
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2 answers
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Is the poset $\mathrm{Idl}_{\neq \emptyset, P}(P)$ of nonempty, proper ideals in a finite connected poset $P$ (empty or) weakly contractible?

$\DeclareMathOperator\Idl{Idl}$Let $P$ be a finite, connected poset with at least two elements, and let $\Idl_{\neq \emptyset, P}(P)$ be the set of downward closed sets $S \subset P$ such that $S \neq ...
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2 answers
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An equality for the reduced homology related to the comparability graph of a poset

$\DeclareMathOperator\width{width}$Let $P$ be a finite poset with $n$ elements (we can assume that $P$ is connected and has width at most $n-2$). The comparability graph $G_P=(V,E)$ associated to $P$ ...
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Another definition of singular homology

The singular homology is defined via standard simplex. Now if I propose another definition of singular homology groups, based on arbitrary simplex, as follows: Let $X$ be a topological space. A $n$-...
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Lifting theorem for finite spaces: replacing perfect normality by normality

In the Lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey, quoted below), can one relax the condition "$A$ is a closed subset of a perfectly normal $X$" to "$A\to X$ has the right ...
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The nerve of the Ising category

A semi-simplicial complex (see Moore's 1958 paper: Semi-simplicial complexes and Postnikov systems, available at http://www-math.mit.edu/~hrm/kansem/moore-semi-simplicial-complexes.pdf, cf MathSciNet) ...
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Unimodality of $f$-vectors of $0/1$-polytopes

It is known that the face vectors (aka $f$-vectors) of general polytopes need not be unimodal. This even fails for simple or simplicial polytopes, as was shown first by Björner. My question is if ...
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Which Stiefel-Whitney numbers can be extended to manifolds with boundaries?

The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-...
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Hamiltonicity for triangulations of the 3-sphere

A classical theorem of Whitney states that the 1-skeleton of every triangulation of the 2-sphere $\mathbb{S}^2$ has a Hamilton cycle as long as each of its 3-cycles bounds a triangle. I'm wondering if ...
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Euler characteristic of pseudomanifolds with boundary

It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that $$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$ In particular, if ...
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Properties a triangulation must have in order to describe a manifold

I am mainly interested in the $3$-dimensional case. It is a well-known fact, following from the work of E. E. Moise and R. H. Bing in the 1950s, that every $3$-dimensional topological manifold (with ...
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Simplicial realization of the circle action on the free loop space

Given a simply connected topological space $X$, it is well known that its free loop space $LX$ has cohomology being the Hochschild homology of the singular cochains [1]: $$HH_\bullet(S^\star X) \simeq ...
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Simplex invariants?

Let $k=s$ be a positive definite symmetric function and a simililarity over the natural numbers such that $k(a,a) = 1 $ for all natural numbers $a$. A similarity $s:X\times X \rightarrow \mathbb{R}$ ...
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1 answer
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An acyclic simplicial complex where all links are generalised homology spheres

We say that a simplicial complex $K$ is acyclic if it's integral reduced simplicial homology groups are trivial in all dimensions. For a vertex ${v} \in K$, we define the link $$lk(v) :=\{\sigma \in K ...
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"Permutation matrix" but non-zero entries are replaced by $e^{ix}$

Birkhoff–von Neumann theorem states that a polytope formed by a set of doubly-stochastic matrices has extreme points that are permutation matrices. I am wondering if there is a similar theorem for a ...
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3 votes
1 answer
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Is every map from a simplicial complex to a sphere homotopic to a simplicial map to the boundary of a k-simplex?

Suppose that $X$ is a simplicial complex, and $f:X \rightarrow S^k$ a continuous map to a sphere. Is $f$ always homotopic to a simplicial map to the boundary of a $(n+1)$-simplex, $\partial \Delta^k$? ...
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1 answer
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For which pairs $k$ and $n$, $n\mid{{n-2} \choose {k}}$

The question This question that arose in a discussion with Ron Adin is quite simple: For which pairs $k$ and $n$ does $n$ divide ${{n-2} \choose {k}}$? Simple observations It is easy to see that ...
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5 votes
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Self-dual simplicial complexes

A simplicial complex $K$ on a vertex set $[m] = \{1,...,m \}$ - where here we say vertex set to mean an ambient set of vertices on which $K$ is defined - is self-dual if it is equal to its Alexander ...
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1 answer
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Irreducible representations of the symmetric group on homology of simplicial complex

I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects: Consider an action of $\Sigma_k$ on a finite ...
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Enumeration and encoding of simplicial complexes

I'd like to know how to enumerate and encode all (abstract) simplicial complexes of a given kind. To start as simple as possible, consider the familiy $\mathcal{S}_n^{d}$ of simplicial complexes which ...
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7 votes
1 answer
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Are there simplicial spheres with "non-geometric symmetries"?

Let $\Delta$ be a simplicial sphere, that is, a finite (abstract) simplicial complex whose canonical geometric realization $|\Delta|$ is homeomorphic to a sphere $\mathbf S^d\subset\Bbb R^{d+1}$. ...
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8 votes
1 answer
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What is a subdivision of an abstract simplicial complex?

I am looking for the definition of the subdivision of a simplicial complex. When the complex is defined in a geometric way, then the definition is pretty simple : the complex σ(C) is a subdivision of ...
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3 votes
0 answers
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When is it possible to extend several linear orders defined "locally" into a single linear order defined "globally"?

This is a somewhat fuzzy question, so I will try my best to give a formulation which includes everything relevant while excluding everything else. I would like to find out if anyone else has studied ...
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Can a spherical simplicial complex have more than one "central" inversion?

Let $\Delta$ be a finite connected simplicial complex. Call a simplicial map $\phi:\Delta\to\Delta$ an inversion if $\phi$ is an involution, that is $\phi\circ\phi=\mathrm{id}$, and $\phi$ is not ...
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4 votes
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"Baues poset" of shellings of simplicial polytope?

Let me start with some background I want to use as analogy. Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: ...
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1 vote
0 answers
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Generalization of Moore graphs -- not geometries

I was thinking about the following problem: Suppose you have a $k$-uniform hyper graph (simplicial complex of dimension $k$) with a complete $(k-1)$-skeleton, and some form of regularity (e.g. every ...
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  • 433
7 votes
2 answers
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Constructing a $0/1$ polytope from an abstract simplicial complex

Let us fix $\Delta$ a finite simplicial complex, and label the vertices of $\Delta$ as $\{1,2,\ldots,n\}$. For each $F\in \Delta$ let us consider the point in $\mathbb{R}^n$ given by: $$e_F := \sum_{i\...
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  • 1,716
3 votes
2 answers
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Inequality of $h$-vectors of shellable simplicial complexes

I've been studying the article of Bjorner entitled "Homology and shellability of matroid complexes". At a certain point he states an exercise that says: Let $\Delta$ be a shellable ...
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3 votes
3 answers
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Are there characteristic-dependent Betti numbers in characteristic not equal to two?

Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
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3 votes
0 answers
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Skeleton of $\mathcal{G}$-simplicial complex

I'm trying to prove a result with simplicial complex endowed with an action of a groupoid. Let start with a formal definition : $\textbf{Definition.}$ [$\mathcal{G}$-simplicial complex] Let $\mathcal{...
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1 answer
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What is the significance of ear decompositions for non-graphic matroids?

On Wikipedia there is subsection in the article on ear decompositions of graphs titled "Matroids": Now as defined above, the circuits of a matroid can not always be listed to satisfy the ...
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8 votes
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Is there a combinatorial representation of general topological manifolds similar to triangulations?

Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
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5 votes
2 answers
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Does the union of the subcomplexes of $X$ that contain a given subcomplex and whose inclusion in $X$ is trivial on $\pi_1$, have trivial $\pi_1$?

Let $X$ be a simplicial complex and let $A \subset X$ be a contractible subcomplex on the same set of vertices as $X$. Is it true that the union $$\bigcup C$$ taken over all complexes $A \subset C \...
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5 votes
1 answer
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Existence of equivariant triangulations

I'm looking for a compatibility result which links two types of structures that could be imposed on a topological space $X$: Call $X$ triangulable if there exists a finite simplicial complex $K$ ...
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11 votes
1 answer
224 views

A class of simplicial complexes defined by arithmetic properties

The purpose of my question is to ask about properties in a certain class of 3-dimensional (and other odd dimensional) simplicial complexes. I will first describe the construction in 3 dimension and ...
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3 votes
0 answers
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On a generalization of the Borsuk-Ulam theorem / Tucker's lemma for a map from simplex to its boundary

The Borsuk-Ulam theorem is equivalent to $S^{n-1}$ not being a retract of $B^n$. <totally wrong! or else 2+2=4 is equivalent to the Poincare conjecture/thm> How shall i prove the following ...
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3 votes
0 answers
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F-vectors of simplicial complex and f-vectors of non-faces of simplicial complex

Is there any result which gives us a relation between f-vector of simplicial complex and f-vector of nonfaces of a simplicial complex? Thank you
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7 votes
1 answer
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Visualize (co)sketeton of a simplicial set (geometrical intuition)

I want to understand if there is an intuition approchable with most possible 'elementary geometrical' knowledge for $n$-(co)skeleta of simplicial sets? Formally sketleton & coskeleton functions ...
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2 votes
0 answers
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Combinatorial condition for orientability on simplicial complexes

Let $K$ be a simplicial complex whose geometric realization is a topological or smooth manifold. Is it possible to restate the condition of orientability of $M$ exclusively in (combinatorial) terms of ...
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2 votes
3 answers
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Alternating sum over collections of sets

Let $\mathbf{P}$ be a collection of subsets of a finite set $X$. Let $\mathscr{S}$ be the set of all subsets $\mathbf{S}\subset \mathbf{P}$ such that $\bigcup_{S\in \mathbf{S}} S = X$. Can one give a ...
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3 votes
0 answers
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Acyclic extensions of acyclic simplicial complexes

Say an abstract simplicial complex $X$ is acyclic if its reduced integral simplicial homology groups $\tilde{\mathrm{H}}^{\Delta}_p(X)$ vanish for all $p\geq 0$. Is it the case that, for any $n>0$, ...
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9 votes
2 answers
371 views

Alternating sum over collections closed under containment

Let $\mathscr{C}$ be a collection of subsets of a finite set $P$. Assume $\mathscr{C}$ is closed under containment: if $S\subset P$ is in $\mathscr{C}$, then every set $S'\subset P$ containing $S$ is ...
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6 votes
1 answer
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Subdivision of closed homology manifold reference request

I am interested in the barycentric subdivision of closed homology manifolds. Definition A (finite) simplicial complex $K$ is a closed homology manifold of dimension $n$ if for every $k$-simplex, its ...
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8 votes
0 answers
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Small flag triangulations

In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...
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