All Questions
10 questions
8
votes
1
answer
498
views
Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
From "The multiple facets of the associahedra" by Loday:
Let us consider the formal power series
$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$
and let
$$ g(x) = x+b_1 x^2 + ...
- 10.5k
7
votes
0
answers
281
views
Relations between Betti numbers for clique complex
Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ ...
- 8,071
6
votes
1
answer
812
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 ...
- 385
6
votes
1
answer
479
views
Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?
I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...
- 6,937
6
votes
1
answer
530
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 ...
4
votes
1
answer
362
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. ...
- 1,500
2
votes
1
answer
143
views
Triangles and convex hulls in high dimensions
Given a set $S_n$ of $n$ points $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\in\mathbb{R}^d$, such that every $(d+1)$-tuple in $S_n$ is affinely independent, and let $C(S_n)$ be the convex hull ...
- 1,677
2
votes
1
answer
111
views
Maximum genus of an abstract "cycle complex"
Let us define an abstract "cycle complex" as the following combinatorial object: it is $(V, C)$, where $V$ is a set of $n$ nodes, $C$ is a set of $c$ cyclically ordered subsets of $V$, each ...
- 1,389
1
vote
1
answer
378
views
Bridges between geometry and combinatorics
Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a ...
- 613
1
vote
1
answer
169
views
A proof of Edelstein and Kelly theorem
Edelstein and Kelly theorem states the following.
Let $A$, $B$ and $C$ be $3$ nonempty finite subsets of points in $\mathbb{R}^n$ such that affine-span $(A \cup B \cup C)$ has dimension at least $4$ ...
- 2,576