Questions tagged [simplicial-complexes]

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7 votes
0 answers
128 views

How many simplicial spheres with $n$ vertices and $N$ facets?

Let $s_d(n,N)$ be the number of different $d$-dimensional simplicial spheres on $n$ labelled vertices and $N$ facets (= $d$-simplices). I am in search for the best know upper bounds, especially for $d\...
1 vote
0 answers
85 views

Computing simplicial resolution of rings

As the title says, I would like to ask how we can give "convient" simplicial resolutions of rings. In the category of modules this is often true: ifI have a ring $R$ and some ideal $I$ ...
2 votes
1 answer
96 views

Another lemma on intersections of $d$-simplices

Let $d\ge1$. A $d$-simplex $S$ is the convex hull in $\mathbb R^d$ of the vertices $v_0,\dots,v_d\in\mathbb R^d$ where $\{v_1-v_0,\dots,v_d-v_0\}$ is a linearly independent set of $d$ vectors; for ...
2 votes
1 answer
88 views

A lemma on intersections of $d$-simplices

I have searched in vain for a combinatorial proof of Sperner's Lemma that rigorously proves the following: Let $d\ge0$. A $d$-simplex is the convex hull in $\mathbb R^d$ of the vertices $v_0,\dots,...
4 votes
0 answers
118 views

Triangulating piecewise-linear manifolds

Question 1: Is this the mainstream definition of a PL-manifold? Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
1 vote
0 answers
68 views

Construct manifold given simplical complex

It's known that, in general, given a simplical complex, answering if it's homeomorphic to a manifold is undecidable. However, given a positive answer to the problem, is there an algorithm to construct ...
1 vote
1 answer
85 views

Simplicial cochain representing the pullback of a class Poincaré dual of a submanifold

Let $K$ be a simplicial complex of dimension $n$, $M$ be a topological manifold, and $f \colon |K| \to M$ be a continuous map. Let $X$ be an embedded manifold in $M$ of codimension $n$, such that $f(|...
10 votes
2 answers
429 views

Higher-dimensional Fáry's theorem?

Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges. For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex ...
20 votes
2 answers
866 views

Is there an analogue of the Erdős–Gallai theorem for simplicial complexes?

The Erdős–Gallai theorem gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph. In particular $d_1 \ge d_2 \ge \dots \ge d_n$ ...
6 votes
1 answer
327 views

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 $...
2 votes
0 answers
348 views

What is the nerve of this category?

If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
7 votes
4 answers
949 views

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-...
3 votes
2 answers
208 views

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). ...
3 votes
0 answers
125 views

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 ...
4 votes
1 answer
280 views

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 ...
1 vote
0 answers
80 views

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,\...
2 votes
4 answers
401 views

Number of free faces given n 0-simplexes

Here is my question: How to construct a simplicial complex with $n$ 0-simplex which has the maximum number of free faces? Is there any research topic about this? And is there any relationship between ...
0 votes
1 answer
138 views

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 ...
5 votes
2 answers
338 views

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 ...
6 votes
1 answer
395 views

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 ...
4 votes
1 answer
397 views

Contiguity for simplicial maps between simplicial sets

I begin by recalling the definition of contiguous simplicial maps between abstract simplicial complexes: Definition. Two simplicial maps $\varphi,\psi\colon K \to L$ are said to be contiguous if for ...
4 votes
1 answer
211 views

Nerve theorem for locally infinite covers by subcomplexes

Let $Y$ be a simplicial complex and let $\{Y_i\}_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup_{i\in I}Y_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each ...
5 votes
3 answers
530 views

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 $K$. ...
3 votes
1 answer
150 views

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 $n$...
1 vote
0 answers
42 views

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 ...
4 votes
0 answers
259 views

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 $cd\in E(K_3)$ ...
1 vote
2 answers
152 views

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, ...
2 votes
1 answer
137 views

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 ...
5 votes
2 answers
319 views

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 $...
1 vote
0 answers
34 views

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 ...
15 votes
2 answers
826 views

Turning simplicial complexes into simplicial sets without ordering the vertices

Given an abstract simplicial complex $K$, one can make a simplicial set $X(K)$ with $n$-simplices given by sequences $(x_0, \dotsc, x_n)$ such that $\{x_0, x_1, \dotsc, x_n\}$ is a simplex of $K$. The ...
4 votes
1 answer
207 views

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 ...
2 votes
0 answers
126 views

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 ...
5 votes
1 answer
337 views

"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 $\...
11 votes
1 answer
439 views

How much smaller is the Čech complex than the Vietoris-Rips complex?

The Čech complex is a subcomplex of the Vietoris-Rips complex. The V-R complex includes as a simplex a set of points with pairwise distances at most $\epsilon$, whereas the Č complex includes as a ...
1 vote
0 answers
135 views

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 ...
0 votes
1 answer
309 views

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 ...
5 votes
1 answer
166 views

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}...
2 votes
0 answers
80 views

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 ...
0 votes
0 answers
76 views

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 ...
1 vote
0 answers
69 views

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 ...
4 votes
1 answer
103 views

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 ...
3 votes
0 answers
87 views

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 ...
4 votes
0 answers
167 views

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 ...
1 vote
1 answer
118 views

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, ...
1 vote
1 answer
141 views

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 ...
95 votes
4 answers
10k views

Which manifolds are homeomorphic to simplicial complexes?

This question is only motivated by curiosity; I don't know a lot about manifold topology. Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The ...
41 votes
3 answers
3k views

Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems. Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ ...
8 votes
1 answer
396 views

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 &...
3 votes
0 answers
142 views

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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