Questions tagged [combinatorial-topology]
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19
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Posets whose homotopy type can be efficiently studied without fibrant replacement?
Let $P$ be a poset and $NP$ its nerve. In order to study the homotopy type of $NP$ via the tools of simplicial homotopy theory, we generally need to take a Kan-fibrant replacement of $NP$, e.g. by ...
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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 ...
- 2,769
3
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125
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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 ...
- 2,769
3
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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-...
- 2,769
8
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1
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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 ...
- 81
6
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1
answer
109
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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 ...
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4
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1
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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....
- 2,769
6
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1
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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 ...
- 165
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132
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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 $> 3$...
- 1,832
4
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1
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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.
...
- 1,189
7
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1
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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?
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- 1,189
8
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1
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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 ...
0
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0
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132
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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 ...
- 101
7
votes
1
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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 ...
- 153
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104
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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....
- 170
1
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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 ...
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3
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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.
...
- 147k
25
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5
answers
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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 ...
- 15.2k
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0
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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)...
- 17.9k