Questions tagged [combinatorial-topology]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
31 views

Are there local maps of simplicial (co-)cycles on $d$-manifolds beyond cohomology operations?

I'm interested in locally defined maps of cocycles/chains on manifolds of a fixed dimension $d$ which are compatible with cohomology. To be concrete about what "local" means, let me consider ...
user avatar
  • 2,587
3 votes
0 answers
107 views

Combinatorial fiber bundles

Triangulations (as simplicial complexes) and bi-stellar flips are a combinatorial analogue of (piece-wise linear) topological manifolds. I'm looking for a similar combinatorial analogue for fiber ...
user avatar
  • 2,587
2 votes
0 answers
104 views

Which Stiefel-Whitney numbers can be extended to manifolds with boundaries?

The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-...
user avatar
  • 2,587
8 votes
1 answer
509 views

What is a subdivision of an abstract simplicial complex?

I am looking for the definition of the subdivision of a simplicial complex. When the complex is defined in a geometric way, then the definition is pretty simple : the complex σ(C) is a subdivision of ...
user avatar
6 votes
1 answer
103 views

Subdivision of closed homology manifold reference request

I am interested in the barycentric subdivision of closed homology manifolds. Definition A (finite) simplicial complex $K$ is a closed homology manifold of dimension $n$ if for every $k$-simplex, its ...
user avatar
  • 869
4 votes
1 answer
161 views

What is a sufficient set of links in a simplicial complex to represent any PL manifold?

The link of a vertex in a $n$-dimensional simplicial complex is the $(n-1)$-dimensional simplicial complex formed by the $(n-1)$-simplices each of which, together with the vertex, spans an $n$-simplex....
user avatar
  • 2,587
6 votes
1 answer
385 views

Are triangulations of compact manifolds PL homeomorphic?

I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes ...
user avatar
12 votes
0 answers
129 views

Finite list of neighborhoods to approximate any finite simplicial complex

It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...
user avatar
4 votes
1 answer
232 views

Understanding proof about chromatic number

Consider an undirected graph $K(n,k,i)$, with the all $k$-element subsets of $\{1,\dots,n\}$ as vertices, and two vertices connected by an edge if their sets intersect in less than $i$ elements. ...
user avatar
  • 1,179
7 votes
1 answer
356 views

Kneser graph with overlap

Consider a graph with the vertices being all subsets of size $n$ of a set of size $2n$. Two vertices are connected if their overlap has size at most one. What is the chromatic number of this graph? ...
user avatar
  • 1,179
8 votes
1 answer
346 views

Uniqueness theorem for conformal mapping

Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$. Assume that each has only finitely many ...
user avatar
0 votes
0 answers
128 views

Number of polyhedra with N faces?

A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...
user avatar
7 votes
1 answer
230 views

Existence of a particular kind of polygonal subdivisions of surfaces

Let $\Sigma$ be a closed, compact, connected, oriented smooth $2$-manifold (in other words, a sphere or a torus with $g$ handles). We may draw smooth arcs on the surface in such a way that they cut it ...
user avatar
2 votes
0 answers
102 views

Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear inequalities....
user avatar
  • 170
1 vote
1 answer
85 views

property R_\lambda

A space X has property $R_\lambda$ if every family of its clopen sets of cardinality $\lambda$ has a subfamily of cardinality $\lambda$ which is either linked or disjoint. The following result was ...
user avatar
10 votes
3 answers
976 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. ...
user avatar
25 votes
5 answers
2k views

Testing simplicial complexes for shellability

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consider anything with smaller ...
user avatar
  • 14.7k
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)...
user avatar
  • 17.5k