# Questions tagged [simplicial-complexes]

The simplicial-complexes tag has no usage guidance.

257
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### Signed measures and poset inequalities

Consider a triangulated ball $D$, and assume that $\omega$ is an assignment of real weights
to the simplices of $D$, including the empty one, such that for every maximal simplex $F$ and every simplex $...

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### Hochschild cohomology of path algebra as a cohomology of simplicial complex

M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...

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### Amending flawed "proof" that homology groups are zero

I am trying to prove a certain statement that seems true based on computational data, and there is a nice argument that proves it, assuming all cycles are the simplest ones (e.g., when the only 1-...

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### On algebraic topology of coset complexes without geometry

I'm interested in understanding the algebraic topology of "coset complexes" from a "combinatorial" perspective (i.e., without relying on geometric realizations of the complexes). ...

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### Does Kalai's $3^d$ conjecture hold for simplicial spheres?

Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...

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### A face and all its neighbors: terminology?

Suppose $F$ is a face of a 2-complex, and $F_1,\dotsc,F_n$ are the faces that are adjacent to (i.e., share an edge with) $F$. Is there a standard term for a collection of faces of the form $\{F,F_1,\...

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### Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?

Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:
The facet complex of any simplicial ...

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### Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?

I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...

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### Simplicial set from all orderings of simplicial complex

Given an abstract simplicial complex $K$ on a set of vertices $V$, we can form a semi-simplicial set by $F(K)$ sending $F(K)_n$ to be the set of ordered $(n+1)$-tuples of vertices in $V$ forming an $n$...

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### If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?

Consider a simplicial complex $K$. A piecewise linear map $f: K \to \mathbb{R}^n$ is an almost-embedding if $f(\sigma) \cap f(\tau) = \emptyset$ for any two disjoint simplices $\sigma,\tau$ in $K$.
...

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### Does every triangulable manifold have a vertex-transitive triangulation?

Does every triangulable manifold have a vertex-transitive triangulation?
When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...

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### Homology of infinite matroids of finite rank

Bjorner has a great paper about the homology of independence complexes of finite matroids, which is the usual context in matroid theory as far as I understand. However, I've also been told that often ...

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### Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?

Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...

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### Do there exist smaller simplicial models of barycentric subdivisions?

Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...

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### Lattices formed by unions of elements in an antichain

Let $A_1, \dots, A_k$ be incomparable subsets (of $\{1, \dots, n\}$) and consider the poset $P$ consisting of all possible unions of these under inclusion. Its not hard to see that this is a lattice, ...

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### Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $...

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### A possible cryptomorphism between closure operators and a suitable subclass of simplicial complexes

Good evening to everybody. I'm writing a paper on the combinatorial properties of simplicial complexes and closure operators, but at a certain point I found a problem which seems hard to be solved.
...

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### Decomposition of a simplicial complex into pseudo-manifolds

Is a result of the following type known:
Any finite simplicial complex of dimension $d$ may be, up to collapsing, covered by (not necessarily induced) subcomplexes that are pseudo-manifolds, in the ...

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### Closed good cover of a triangulable space

By a good closed cover of a topological space $X$, I mean a collection of closed subspaces of $X$, such that the interior of them cover $X$, and any finite intersection of these closed subspaces is ...

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### Posets whose homotopy type can be efficiently studied without fibrant replacement?

Let $P$ be a poset and $NP$ its nerve. In order to study the homotopy type of $NP$ via the tools of simplicial homotopy theory, we generally need to take a Kan-fibrant replacement of $NP$, e.g. by ...

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### "Singular homology = simplicial homology" relative to a fibration

Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\...

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### Simplicial sets and oriented simplicial complexes

$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the ...

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### Proving the induced map on the cohomology is an isomorphism

I was going through a paper by Tanaka where I am stuck at the following map "f" which is given by the composition of these maps. Next, he mentions that the induced map is clearly an ...

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### If $X = X_1 \cup \cdots \cup X_n$ is shellable, then is $(X_1 \cup \cdots \cup X_k)\cap X_{k+1}$ shellable?

Let $X = X_1 \cup \cdots X_n$ be a shellable complex, where the $X_i$ are the maximal faces, in the shelling order.
Question 1:
Let $0 \leq k \leq n-1$. Then is $(X_1 \cup \cdots \cup X_k) \cap X_{k+1}...

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### Chain complex of the Salvetti complex of an Artin group

Let $A_\Gamma$ be an Artin group. The Salvetti complex $Sal(A_{\Gamma})$ can be briefly defined as the $2$-presentation complex associated to the usual presentation of the Artin group after attaching ...

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### Topology of independence set of a vector space

This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...

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### Reference Request: Cech cohomology of complexes on an arbitrary site

I am looking for a reference which is equivalent to this stacks project page [1], except formulated in the generality of an arbitrary site. I checked the "Cohomology on Sites" section of the ...

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### A neighborhood of a 2-disc $D\subset\Bbb R^4$ that tapers off towards the boundary?

I am given a PL 2-disc $D\subset\Bbb R^4$ (everything PL from here on) and I need a "neighborhood" $N\simeq B^4$ (PL-homeomorphic to a 4-ball) so that $\partial N\cap D=\partial D$.
If I got ...

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### Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?

Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...

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### In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?

I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...

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### Connectivity of a matroid is at least its rank?

The connectivity $\eta(X)$ of a simplicial complex $X$ is defined as the
$$1+\min_j\{j \mid \tilde{H}_j(X)\neq 0\}.$$
If no such $j$ exists, then $\eta(X):=\infty$.
(See here for this definition, ...

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### Barycentric subdivision and 1-coskeletalization

Let
$sd : sSet \to sSet$ denote barycentric subdivsion;
$cosk_1 : sSet \to sSet$ denote 1-coskeletalization.
Question: Let $X$ be a graph or simplicial set. If the homotopy type of $cosk_1(X)$ is ...

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### Contractible subcomplex containing 1-skeleton?

Question: If $X$ is a simplicial complex that's simply connected and $2$-dimensional, does there always exist a contractible subcomplex $Y$ satisfying $X^{(1)} \subseteq Y$?
The statement is true &...

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### Confused about the proof of a lemma about deleted products

I am confused about the proof of Lemma 2.1 in the paper Obstructions to the imbedding of a complex in a euclidean space. I: The first obstruction, by A. Shapiro, Ann. Math. 66 (1957).
Let $K$ be a ...

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### Homology groups of a certain simplicial complex

I've run across a simplicial complex which, according to Sage, seems to have a very easily-described homology. However, proving this fact has been rather difficult.
Fix $s\ge 2$ (though I would be ...

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### Given a group action on a simplex, can I always find a fundamental region that is a simplex?

Let $\Delta\subset\Bbb R^n$ be a simplex with $n+1$ vertices. Let $G\subset\mathrm{GL}(\Bbb R^n)$ be a finite group of linear symmetries of $\Delta$, i.e. linear transformations that fix the simplex ...

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### Does combinatorial deleted product become equivalent to the topological deleted product after enough subdivision?

Suppose $X$ is a topological space. Define the (topological) $n$-fold deleted product of $X$ to be the space or ordered $n$-tuples of pairwise distinct points in $X$.
$$F(X, n):= \{(x_1, \ldots, x_n)\...

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### Do combinatorially equivalent polytopes have the same triangulations?

A triangulation of a convex polytope $P\subset\Bbb R^n$ is a partition of $P$ into $n$-simplices $\{\Delta_1,...,\Delta_m\}$ each of which has all its vertices among the vertices of $P$. A polytope ...

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### Is every simplicial $d$-sphere linearly embeddable in $\Bbb R^{d+1}$?

A simplicial $d$-sphere is a simplicial complex homeomorphic to the $d$-sphere. It is known that not every such complex can be embedded into $\Bbb R^{d+1}$ as the boundary complex of a convex ...

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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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### 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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### 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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### 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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### 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 ...