# Tagged Questions

**2**

votes

**2**answers

111 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

**1**answer

1k 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

**0**answers

122 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 ...

**34**

votes

**2**answers

898 views

### Is there an analog of Sperner's lemma for the Hopf invariant?

Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic."
My question is, does there exist ...

**2**

votes

**1**answer

144 views

### Non-simplicial triangulations of compact surfaces with few vertices

I'm interested in triangulations with few vertices of a given orientable compact surface $S$.
By triangulation, I don't mean a "simplicial triangulation" but a "decomposition of $S$ by ...

**16**

votes

**1**answer

572 views

### Combinatorial spin structures

I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to ...

**6**

votes

**0**answers

211 views

### What are the homological properties of Young's lattice?

Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. ...

**4**

votes

**1**answer

113 views

### Reference for the image of the adjoint to the differential in graph cohomology (which yields STU & IHX)?

One can define cochain complexes of (combinatorial) graphs, where each term is a vector space of linear combinations of certain (isomorphism classes of) graphs, and where the differential $d$ is a ...

**4**

votes

**2**answers

320 views

### Removing a simplicial subset from a simplicial set

Let $A, X$ be simplicial sets, and suppose there's an inclusion $A \longrightarrow X$. Geometrically realizing the inclusion map, we get a pair of spaces $(\mathcal{A}, \mathcal{X})$. I want to find ...

**3**

votes

**1**answer

338 views

### Any map of a contractible complex to itself has a fixed point

Reading Lovasz's lecture notes on evasive graph properties, I encountered the following extension of Brouwer's fixed point theorem:
Any continues map from a contractible [finite] simplicial complex ...

**4**

votes

**1**answer

407 views

### Triangulation of fundamental domains for surfaces and generators

Sorry, title probably not great, but I can't think of a better one. Here's my question. Suppose I have a $2n$ gon, with the edges identified in pairs, and no neighboring edges identified. Suppose ...

**1**

vote

**0**answers

315 views

### Covering of the torus with simply connected open sets

I know this might sound like standard qualifying exam exercise in algebraic topology, so I apologize from start if this post might be inappropriate. The thing is, I've been doing some elementary ...

**8**

votes

**4**answers

711 views

### References for Eilenberg-Zilber shuffle product

Most of the treatments I can find in the literature for the Eilenberg-Zilber shuffle product approach it from the point of view of simplicial sets (including the original Eilenberg-MacLane paper). I ...

**8**

votes

**0**answers

234 views

### Chern Classes of Exterior Products of a vector bundle.

This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of ...

**8**

votes

**2**answers

456 views

### Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...

**6**

votes

**3**answers

476 views

### Cylinders dividing $\mathbb{R}^{3}$

Consider $n$ affine copies of a compact cylinder, say $S^{1}\times [-3,3]$ with top and botom, sitting inside $\mathbb{R}^{3}$.
For each $n$ we may ask ourselves how to arrange the $n$ cylinders so ...

**2**

votes

**2**answers

205 views

### product of two sub-Grassmannians

Let $G(k,n)$ be the Grassmannian of complex $k$-planes in $\mathbb{C}^n$. Then for $k_1+k_2=k$ and $n_1+n_2=n$, $G(k_1,n_1)\times G(k_2,n_2)$ is a submanifold of $G(k,n)$. So the cohomology class of ...

**12**

votes

**3**answers

798 views

### How do facts about the homotopy type of cell complexes shed light on analytic number theory?

I just saw this link text interesting MO question, with a link to this paper, which uses facts from the topology of cell complexes to derive facts of an analytic number theory flavor.
From the ...

**10**

votes

**3**answers

536 views

### Acyclic categories related to structures in algebraic topology

An acyclic category (also called loopfree category or scwol (small category without loops)) is a small category where only identity morphisms have inverses, and any morphism from an object to itself ...

**15**

votes

**1**answer

459 views

### How many simplicial complexes on n vertices up to homotopy equivalence?

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is ...

**3**

votes

**2**answers

272 views

### Discrete version of some topological object.

Consider a triangulated orientable surface with the following data: on each edge a vector with integer coordinates is written so that for each triangle the sum of the vectors corresponding to three ...

**5**

votes

**1**answer

364 views

### Terminology Concerning Oriented Simplicial Complexes

An oriented simplicial complex is a simplicial complex K equipped with a partial ordering on its vertices that restricts to a linear ordering on each simplex. I am wondering if there is a standard ...

**23**

votes

**1**answer

895 views

### High Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead d=5,4). Edit: I mistakingly identified my mistake, it is 6,5 but I got the indices shifted by one.]
Background: Polygon spaces
Given a ...

**13**

votes

**0**answers

772 views

### Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?

**15**

votes

**1**answer

584 views

### Calculating Mayer-Vietoris efficiently

This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book ...

**6**

votes

**2**answers

430 views

### A flag complex is contractible iff the underlying graph is…?

Let $G$ be a finite simple graph and let $C(G)$ be the flag complex associated to $G$ (the set of vertices of $C(G)$ is the vertex set of $G$ and the set of all cliques of $G$ are its simplexes).
...

**3**

votes

**0**answers

231 views

### Coloring $\mathbb{Z}^k$ and a fixed point theorem

This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...

**2**

votes

**2**answers

258 views

### algorithms for comparing two simplicial complexes

Given a set $A$ of subsets of $\{1, \ldots n\}$ which is closed under taking subsets, let $X(A)$ be the corresponding simplicial complex, i.e. simplices of $X(A)$ are elements of the set $\bar A$, and ...

**4**

votes

**2**answers

576 views

### Presentation of the pure Artin groups

Let $W$ be a Coxeter group attached to a Coxeter matrix with entries $m_{ij}$ . The presentation of $W$ is given by
$$W= < T_1, \dots, T_n | T_i^2=1, T_iT_jT_i \ldots = T_jT_iT_j \ldots, i \neq ...

**1**

vote

**1**answer

596 views

### Applications of ham sandwich type results. References? A general principle?

Lately there has been a lot of interest on applications of the ham sandwich theorem and related results. There is a bunch of lecture notes and surveys that touch upon the subject. I dont know of any ...

**7**

votes

**3**answers

393 views

### Equilibrium configurations of ions on n-Dim balls.

Given an n-dimensional electrically neutral, solid metal ball (a point for n=0; a rod, n=1; a disc, n=2; a solid ball, n=3; ...), place N=(n+1)! identical ions on the ball. As one of my favorite ...

**8**

votes

**1**answer

694 views

### A Desirable Extension of the Nerve Theorem

Backgroud
The Nerve Theorem (see nLab;) asserts that given a finite collection $\cal K$ of compact sets with the property that all non empty intersections of sets in the family are homotopically ...

**18**

votes

**0**answers

2k views

### Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...

**6**

votes

**2**answers

1k views

### mnev's universality corollaries, quantitative versions?

Mnev's universality theorem claims that any semialgebraic set is the realization space of some oriented matroid. Moreover, the rank of the or matroid can be prescribed in advance.
1.-Are there ...

**7**

votes

**2**answers

292 views

### non-rigidity of interior points in polyhedral triangulations?

It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into ...

**13**

votes

**1**answer

486 views

### Homeomorphism type of union of two balls intersecting in a ball.

Let $d$ be an integer. Let $A,B \subseteq \mathbb R^d$ be two sets homeomorphic to an open $d$-ball such that their intersection is again homeomorphic to an open $d$-ball. Does it follow that their ...

**21**

votes

**8**answers

1k views

### Avatars of the ring of symmetric polynomials

I'm collecting different apparently unrelated ways in which the ring (or rather Hopf algebra with $\langle,\rangle$) of symmetric functions $Z[e_1,e_2,\ldots]$ turns up (for a Lie groups course I will ...

**12**

votes

**3**answers

449 views

### Can we map every graph in the plane such that all induced cycles selfintersect?

Suppose we have a graph G. Is it true that we can map its vertices to the plane such that when connecting neighboring vertices with segments, then any induced cycle of G that has length at least 4 ...

**3**

votes

**2**answers

302 views

### Poset fiber theorems under a special assumption on the poset map?!

Hey everyone, I am facing the following problem:
Say that a (order-preserving) poset map $f:P\to Q$ has property $(\star)$ if for all $q_1,q_2\in Q$ with $q_1\leq q_2$ and every $p_2\in f^{-1}(q_2)$ ...

**10**

votes

**4**answers

849 views

### What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true?

What I would like to know is exactly what the title asks:
What are the most general classes of
simplicial complexes or posets for
which the Charney-Davis conjecture is
known, and what is the ...

**4**

votes

**2**answers

489 views

### Discrete Morse theory and existence of minimal complex

A minimal complex is a CW complex whose only cells are the homology cells.
Is there some sort of criterion on CW complexes about existence of minimal complexes?
Actually I am working on a problem ...

**9**

votes

**4**answers

762 views

### Computing homology of very large posets

I'm studying the homology of a couple of very large posets (one has over 4 million vertices, though the dimension is only 3). I want to show the posets are spherical (homology vanishes except in top ...

**3**

votes

**1**answer

286 views

### Posets of finite sequences are highly connected

I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X_{n,k}$ ...

**0**

votes

**1**answer

170 views

### Is the Euler characteristic of a certain nonlinear variety related to that of a certain linear variety?

(This is a generalization of a question I posted a week ago.)
I'm looking at a variety sitting inside the algebraic torus $(\mathbb{C}\setminus 0)^n$ generated by the ideal $I = (*x_1^{\alpha_1} + ...

**3**

votes

**1**answer

572 views

### Automorphisms of the rooted tree operad

This follows Ryan Budney's comment to the question asked here.
What is the automorphism group of the rooted tree operad?
(By the rooted tree operad, I just mean the operad with object rooted ...

**1**

vote

**2**answers

685 views

### What are operad automorphisms?

What is the general concept of an *operad automorphism*$?$ Is there a "standard" definition?
[added after comment] If an operad automorphism is an invertible operad endomorphism, how then is operad ...

**18**

votes

**2**answers

958 views

### Graphs, K-theory and combinatorial balls: conjectures

The following conjectures from Kapranov and Saito's Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions aren't as well-known as they aught to be, so I'd like to state ...

**15**

votes

**0**answers

1k views

### Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...

**11**

votes

**2**answers

706 views

### covers of $Z^\infty$

Is it possible to cover $Z^\infty$ (the infinite direct sum of $Z$'s with the $l_1$-metric) by a finite set of collections of subsets $U^0,...,U^n$ such that each collection $U^i$ consists of ...

**4**

votes

**2**answers

418 views

### Intersection homology for toric varieties

is there any algorithm known for computing (middle perversity)intersection homology of complex toric varieties based on their combinatorial data? I'm not looking for a computer program.
Regards,
...