# Tagged Questions

**7**

votes

**2**answers

327 views

### Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...

**5**

votes

**1**answer

273 views

### Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...

**4**

votes

**1**answer

319 views

### Is every finite subcomplex of a contractible simplicial complex contained in a finite contractible subcomplex?

The question is as in the title:
Is every finite subcomplex of a contractible simplicial complex $K$ contained in a finite contractible subcomplex of $K$? What if we are allowed to take ...

**15**

votes

**2**answers

440 views

### If $X$ is a simplicial complex, is their a characterization of the links of the vertices of $X$ that is equivalent to the statement "$|X|$ is a manifold

We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a ...

**1**

vote

**0**answers

103 views

### Do bistellar flips in a simplicial sphere preserve facet connectivity?

Define two facets of a simplicial sphere as k-connected if there are at least k disjoint edge-vertex paths between them in their facet-ridge graphs.
Do bistellar flips preserve facet k-connectivity?
...

**13**

votes

**0**answers

307 views

### Asymptotics for the number of triangulations of a manifold M

In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)
I want to emphasize a problem which
comes from mathematical physics which
is unsolved which is ...

**13**

votes

**1**answer

413 views

### When can a contractible 2-complex be embedded in R^3?

Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of ...

**1**

vote

**1**answer

154 views

### Is there a linear embedding of a simplical 3-complex in R^6?

I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...

**1**

vote

**1**answer

289 views

### Do bistellar flips preserve shellability?

I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip ...

**17**

votes

**2**answers

718 views

### Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...

**15**

votes

**2**answers

1k views

### Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...

**15**

votes

**4**answers

2k views

### Why is complex projective space triangulable?

In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...

**4**

votes

**1**answer

731 views

### Ambiguous definition of “nerve of an open covering” on wikipedia?

Let $(U_i)_{i\in I}$ be an open covering of a topological space $X$.
At http://en.wikipedia.org/wiki/Nerve_of_an_open_covering,
the nerve of the open covering is defined as follows:
the nerve $N$ ...

**7**

votes

**3**answers

757 views

### Triangulations coming from a poset. Or: What conditions are necessary and sufficient for a finite simplicial complex to be the order complex of a poset?

Every partially ordered set gives a triangulation of (the geometric realisation of) its order complex. (The n-simplices of the order complex are the chains $x_0\leq x_1\leq\cdots\leq x_n$.) However, ...

**6**

votes

**2**answers

271 views

### Simplicial and cubical decompositions of low valence

Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...