Questions tagged [simplicial-complexes]
The simplicial-complexes tag has no usage guidance.
31
questions
18
votes
3
answers
1k
views
Are finite spaces a model for finite CW-complexes?
Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, ...
- 42.5k
17
votes
3
answers
2k
views
Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?
What is the best upper bound known on the (absolute value of) the
Euler characteristic of a simplicial complex
in terms of the number of its facets ?
In particular, I am interested in proving or ...
- 173
7
votes
2
answers
1k
views
Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
- 10.3k
23
votes
0
answers
2k
views
Are there lots of integer homology three-spheres?
The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and references)...
- 18.3k
19
votes
2
answers
2k
views
A simplicial complex which is not collapsible, but whose barycentric subdivision is
Does anyone know of a simplicial complex which is not collapsible but whose barycentric subdivision is?
Every collapsible complex is necessarily contractible, and subdivision preserves the ...
- 323
16
votes
2
answers
2k
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 $2$-...
- 7,755
14
votes
1
answer
407
views
Realisation of maps between spheres by simplicial maps
Let $K^n_0 := \partial \Delta_{n+1}$ the simplicial set obtained by removing the $(n+1)$-simplex from the standard simplex. This gives a simplicial decomposition of the sphere $S^n$. More generally, ...
- 9,591
10
votes
1
answer
586
views
Non-triangulable 4-manifold as a boundary of some 5 manifold
We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
- 10.3k
10
votes
3
answers
1k
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.
...
- 149k
10
votes
1
answer
393
views
Shellable simplicial complex with restriction on shellings
Does there exist a (pure) shellable simplicial complex $\Delta$ with the
following property? There is some facet $F$ of $\Delta$ such that no
shelling can begin with $F$.
This condition is easily ...
- 49.5k
10
votes
2
answers
625
views
Seeking very regular $\mathbb Q$-acyclic complexes
This question was raised from a project with Nati Linial and Yuval Peled
We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties
a) $K$ has a complete $...
- 24.2k
9
votes
3
answers
449
views
Minimal combinatorial data needed to define a polytope [duplicate]
Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
- 1,335
9
votes
2
answers
401
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 ...
- 19.4k
8
votes
1
answer
219
views
Extending a triangulation of the boundary of $M \times I$
(Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $M$, I mean a homeomorphism with the geometric realization of a simplicial complex, $h: |K| \...
- 5,319
7
votes
2
answers
707
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 ...
- 71
7
votes
2
answers
662
views
Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!) What can be said about subcomplexes of 2-complexes deformation retractible onto graphs.
I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
Is it true that the Euler characteristic of a finite connected aspherical ...
- 1,462
6
votes
3
answers
277
views
0-1 matrix corresponding to an abstract simplicial complex
Let $A$ be a 0-1 matrix whose columns are maximal. We can associate its rows with vertices and columns with simplices in an abstract simplicial complex. Conversely, given an ASC, we can encode it in a ...
- 15.3k
6
votes
0
answers
239
views
Dowker and neighborhood complexes: reference wanted
Let $R$ be a 0-1 matrix whose rows or columns are maximal.
Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)?
From 0-1 matrix corresponding to an abstract simplicial ...
- 15.3k
6
votes
0
answers
329
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 ...
4
votes
1
answer
607
views
Homology of simplicial complex versus homology of simplicial _set_
Let $K$ be a simplicial complex: it consists from the set (called the set of vertices) and a family of subsets of set of vertices satisfying the property of being closed under taking subsets (those ...
- 9,150
4
votes
0
answers
896
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 $[...
- 41
3
votes
1
answer
670
views
Homotopy equivalence of geometric realizations
This question is related with this one. For simplicial complex (which we have to assume is ordered as explained in the answer of the linked question) we have a construction of geometric realization ...
- 9,150
3
votes
0
answers
120
views
Is the Evasiveness Conjecture strong enough to constructively imply the negation of the Pizzazz-conjecture?
Question. Assume the truth of the (notoriously open) Evasiveness Conjecture. Does this constructively imply the negation of the Pizzazz-conjecture?
Remarks.
The relevant statements are, by ...
- 6,001
3
votes
1
answer
155
views
Cyclic polytopes whose boundary is a flag complex
A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ ...
- 1,017
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 ...
- 1,388
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,...
- 1,388
2
votes
0
answers
357
views
Simplicial complexes are to PL structures of manifolds as simplicial sets are to what?
A simplicial complex is a PL structure for, and thus also 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 ...
- 5,545
2
votes
4
answers
403
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 ...
- 29
2
votes
3
answers
466
views
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 ...
- 19.4k
1
vote
0
answers
148
views
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 ...
- 317
1
vote
1
answer
402
views
Algorithm to check whether simplices intersect nicely
Suppose that $A$ and $B$ are both $3$-simplices linearly embedded in $\mathbb{R}^3$, say with vertices in $\mathbb{Q}^3$ so that we can do computations exactly. (I am also interested in the ...
- 55k