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3
votes
1answer
192 views

Does the fat geometric realization take limits to homotopy limits?

I am deeply confused about geometric realizations and finite limits. Suppose that I am working with simplicial sets (I dont need simplicial spaces) that are "good" in the sense of Segal so that I can ...
2
votes
0answers
88 views

What kinds of complexes can be collapsed to?

A simplicial complex $S$ is collapsible if there is a sequence of elementary collapses that bring $S$ down to a single point; I'll denote this as $S \searrow \{pt\}$. I am wondering about a similar ...
1
vote
1answer
65 views

Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...
2
votes
0answers
141 views

Groups acting on complexes

Let $G$ be a finite group. We define a $G$-simplicial complex $\mathcal{A}(G)$ with set of vertices $G^*:=G-\{e\}$ and the simplices are the abelian subsets of $G^*$. The groupe $G$ act simplicially ...
0
votes
0answers
106 views

Relating overlapping simplicial complexes

Let $X$ be a simplicial complex and let $A,B\subset X$ be subcomplexes such that $C=A\cap B$ is a non-empty simplicial complex. Finally, let $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, ...
1
vote
1answer
73 views

homotopic maps of locally finite spaces

It is well known that if $X,Y$ are $T_0$ Alexandrov spaces then they are just posets. With every such spaces we can associate an abstract simplicial complex $K(X)$ where the simplices are nonempty ...
5
votes
1answer
230 views

Contexts and notations for composing asymmetric simplices

Imagine the elements of a group-like structure as puzzle pieces with essential two sides, an IN-side and an OUT-side. You can compose two such pieces in two obvious ways: Now consider triangular ...
1
vote
1answer
129 views

Does the nerve functor preserve fibrations?

As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ? By fibration of groupoids I mean a fibration for the ...
0
votes
1answer
95 views

contracting homotopy on simplicial sets

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy $$ F\: : \: I\times PM\to PM $$ Defined by $F(s,y)(t):=y(st)$. This map is an ...
6
votes
1answer
210 views

A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint): Taking $A$ to be a point and iterating this collapsing lemma, this ...
2
votes
1answer
138 views

Is barycentric subdivision of a collapsible, regular CW complex collapsible (non-evasive)?

Let $X$ be a finite, regular CW complex, and let $X'$ be its barycentric subdivision (i.e. the order complex of the face poset of $X$). Assume $X$ is collapsible. Is $X'$ collapsible? Is $X'$ ...
2
votes
1answer
137 views

Vanishing homology of simplicial complexes with few facets

Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$. Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j ...
13
votes
2answers
407 views

Flag complexes that are shellable but not vertex decomposable

As the title suggests, I was wondering if anyone can point me to any examples in the literature to flag complexes that are shellable but not vertex decomposable. It is well-known that if a ...
2
votes
0answers
43 views

Any results on rayless simplicial complexes?

We define a closed ray in a topological space $X$ to be its closed subset homeomorphic to the real half-line $[0,\infty)\subseteq \mathbb{R}$. Call a topological space $X$ rayless if it does not ...
4
votes
2answers
174 views

Simplicial complices on unlabelled vertices

My question is about (abstract) simplicial complices. In particular, how many are they if I consider $n$ unlabelled vertices? For example, if $n=4$, the two complices $$ \{\varnothing, \{1\}, \{2\}, ...
3
votes
1answer
195 views

Kan condition in simplicial homotopy theory

I know Kan condition (see http://en.wikipedia.org/wiki/Kan_fibration) is something like homotopy extension condition, and I know this condition ensures homotopy defined by the naive idea to be an ...
15
votes
0answers
175 views

Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...
8
votes
0answers
196 views

Consequences of Zeeman's conjecture

Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision. Zeeman showed that this implies the Poincaré conjecture in ...
2
votes
0answers
63 views

Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?

Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | ...
2
votes
0answers
86 views

A natural simplicial object in the simplicial category (?)

In several works (es. [CS]) the study of the properties of the simplicial category $\Delta$ reveals fundamental aspects of universal properties (eg monoid generator) or basic constructions (eg ...
2
votes
2answers
149 views

Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$. Now, if $G$ is a well-covered graph (where all maximal ...
14
votes
1answer
2k views

What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...) Are there examples of results in "classical" [*] graph theory that have been achieved by using simplicial ...
5
votes
0answers
181 views

Is Euler-characteristic of a simplicial complex on $n$ vertices and $f$ facets at most $n^{O(\log f)}$?

(Definition: Facet = Maximal Face) This question is a continuation of the previous one that I had asked a couple of years ago: Is Euler characteristic of a simplicial complex upper bounded by a ...
11
votes
0answers
438 views

Has this chain complex associated with a simplicial complex been studied before?

I have stumbled upon a construction that has probably been noticed before, and I wonder if anyone can point me to a reference. Suppose that $K$ is a simplicial complex. Let $P(K)$ be the free abelian ...
5
votes
1answer
152 views

References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other. This ...
3
votes
1answer
90 views

Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...
1
vote
0answers
165 views

fixed simplicial complex under group action

I have found in an article dealing with combinatorial manifolds the following definition: Let $C$ be a finite simplicial complex, and let $G$ be a finite group acting by automorphisms of $C$. The ...
2
votes
0answers
236 views

When is the realisation of a simplicial set a manifold?

It is known that a simplicial complex is homeomorphic to a manifold if the link of every vertex is a simplicial sphere, for which there exists a definition. (I know that for high dimension, the ...
1
vote
1answer
190 views

Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...
3
votes
1answer
293 views

What is the “higher version” of chain homotopy in singular homology?

In basic algebraic topology, we know the following well-known chain homotopy theorem: Let $X$ be a topological space and $I=[0,1]$ be the unit interval. Let $S_*(X)$ and $S_*(X\times I)$ be the ...
-1
votes
1answer
110 views

Poset complex of reverse ordering [closed]

This might be too easy but I cannot proof it easily. Any reference or hint will be great. Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. ...
1
vote
1answer
173 views

3-complexes not embeddable in 3-space

My question is about embeddability of 3-dimensional complexes in R^3. Do we have something like Kuratowski's theorem for complexes in 3-space which specifies a set of minors for non-embeddability?
5
votes
1answer
143 views

Injective simplicial maps between Arc complexes

Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
3
votes
1answer
155 views

Lipschitz Approximation to a PW Smooth Map

Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...
3
votes
1answer
144 views

Thinking about the quadratic dual graphically

Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed ...
1
vote
1answer
199 views

simplicial complex equipped with barycenric metric is complete [closed]

Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a ...
11
votes
2answers
648 views

How many triangulations of the genus $g$ surface on $n$ vertices?

By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations $t(n)$ of the ...
1
vote
0answers
91 views

Is this basis of simplex polynomials known?

Put $R_n=\mathbb{R}[t_0,\dotsc,t_n]/(\sum_it_i-1)$ (the ring of polynomial functions on the $n$-simplex). Consider a monomial $t^a=t_0^{a_0}\dotsb t_n^{a_n}$. Let $(b_0,\dotsc,b_n)$ be the sequence ...
6
votes
1answer
321 views

Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...
2
votes
3answers
571 views

Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...
3
votes
0answers
263 views

Simplicial chain complex with ordered simplices

Let $X$ be an abstract simplicial complex. Recall that the usual simplicial chain complex for $X$ is defined as follows. Let $C_k(X)$ be the quotient of the free abelian group on formal symbols ...
7
votes
2answers
422 views

Combinatorial distance between simplicial complexes

Let $K_1$ and $K_2$ be two simplicial complexes. I am seeking a measure of the distance between $K_1$ and $K_2$ when viewed as combinatorial objects. What I have in mind is something like this. ...
1
vote
0answers
146 views

Cohomology with compact support and the nerve of a recouvrement

Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...
3
votes
1answer
250 views

Are there “geometrically nice” sets from which to construct coverings that admit “Vietoris-Rips like” approximations to the nerve?

It is well known that the nerve (or Čech complex) of a covering consisting of metric balls with a common fixed radius is nicely approximated by the Vietoris-Rips complex. Being a flag complex on its ...
7
votes
2answers
270 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 ...
26
votes
3answers
1k 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 $K$ ...
4
votes
0answers
169 views

When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where ...
17
votes
2answers
528 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 $d_1 \ge d_2 \ge \dots \ge d_n$ ...
53
votes
0answers
3k views

The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
2
votes
1answer
140 views

coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...