1
vote
0answers
172 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
4
votes
1answer
230 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
7
votes
0answers
229 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
2
votes
0answers
320 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
2
votes
2answers
132 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
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
0answers
155 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 ...
36
votes
2answers
925 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
1answer
154 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 ...
15
votes
1answer
639 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 ...
7
votes
0answers
230 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
1answer
121 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
2answers
324 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
1answer
354 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
1answer
413 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
0answers
355 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
4answers
768 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
0answers
269 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
2answers
468 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
3answers
491 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
2answers
215 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
3answers
812 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
3answers
560 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
1answer
497 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
2answers
280 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
1answer
426 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 ...
26
votes
1answer
1k views

High-Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.] Background: Polygon spaces Given a ...
15
votes
0answers
884 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
1answer
616 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
2answers
445 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
0answers
235 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
2answers
260 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
2answers
658 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
1answer
624 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
3answers
397 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 ...
9
votes
1answer
735 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 ...
17
votes
0answers
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
2answers
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
2answers
295 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
1answer
496 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
8answers
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
3answers
460 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
2answers
323 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
4answers
897 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
2answers
594 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
4answers
777 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 ...
4
votes
1answer
294 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
1answer
171 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
1answer
583 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
2answers
688 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 ...