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6
votes
0answers
126 views

What is known about the $q$-analogue of the simplex?

I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...
2
votes
1answer
68 views

How a “sequentially Cohen–Macaulay” simplicial complex relates to “Cohen–Macaulay” simplicial complex?

Let $\Delta$ be a simplicial complex on $[n]$ of dimension $d − 1.$ Let $0\le i\le d-1.$ One defines the pure i_th skeleton of $Δ$ to be the pure subcomplex $\Delta(i)$ of $\Delta$ whose facets are ...
8
votes
0answers
117 views

Computer searches for the $g$-conjecture

McMullen's $g$-conjecture aims the classify possible $f$-vectors of simplicial $d$-spheres. The $g$-conjecture has been proven for polytopal spheres and for simplicial spheres of dimension $d < 5$. ...
6
votes
1answer
110 views

Quotient of Coxeter complex in terms of double cosets?

In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think ...
3
votes
1answer
69 views

relate shellability of a simplicial complex to the links of its faces

Reisner's criterion give a complete characterization of Cohen–Macaulay simplicial complexes, based on $link$s of faces of the simplicial complex. Is there a known fact that relate shellability of a ...
4
votes
1answer
130 views

Equivariant maps from simplicial complexes to spheres

Given a topological space $X$ with involution $\nu$, the $\mathbb Z_2$-index $\text{ind}(X)$ is the minimum integer $n$ such that there exists a map $f:X \to S^n$ which is equivariant with respect to ...
4
votes
1answer
257 views

Who first considered constructibility of simplicial complexes?

A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. ...
11
votes
0answers
103 views

Is every simply connected finite complex the classifying space of a finite monoid

On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...
1
vote
0answers
52 views

if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$

Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual. Let in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$? Is there a ...
4
votes
1answer
149 views

Generalized ordering on simplicial complex

The vertices of simplicial complexes are usually totally ordered so that face maps of each simplex can be defined easily for the purposes of homology. That gives an "oriented" simplicial complex. But ...
4
votes
1answer
273 views

What are the 4 convex simplicial 4-polytopes that have 6 vertices?

In Convex polytopes and related complexes by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$. I was wondering what the four $4$-...
7
votes
3answers
433 views

Any PL-homology-manifold is homotopy equivalent to a manifold

Is it true that any compact piecewise linear homology manifold is homotopically equivalent to a (smooth?) manifold of the same dimension? Let me say bit more since my question was wrongly ...
8
votes
1answer
458 views

Mathematics of Chiral Rings

Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex. We now construct $C(A)$, ...
3
votes
1answer
65 views

Billera Tree Space

I am studying the tree space of Billera and I do not really understand why it is an Hadamard Space. I have already read L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, ...
2
votes
0answers
51 views

Stiefel manifolds and “simplicial complex chromated Sitefel manifolds”

Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds $$ V_K(\mathbb{R}^n):=\{(v_1,v_2,\cdots,v_k)\in\prod_k\...
3
votes
1answer
246 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
97 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
84 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
150 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
112 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 $C_{\cdot}(X)$, $C_{\cdot}(A)$, $C_{\cdot}(B)$, $C_{\cdot}...
1
vote
1answer
78 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 ...
13
votes
2answers
454 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
163 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
147 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
238 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
184 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
147 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 \...
14
votes
2answers
509 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
49 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
185 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\}, \...
4
votes
1answer
219 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
217 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, ...
9
votes
0answers
220 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
78 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} | \{x_{i_1},\ldots,x_{i_{r}}\}\not\...
2
votes
0answers
88 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
158 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 ...
6
votes
0answers
194 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 ...
12
votes
0answers
468 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
198 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
141 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
236 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
247 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
195 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
313 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
133 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
180 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
152 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
165 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
156 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 $\mathbb{Z}...