# Tag Info

Accepted

### Has anyone ever actually seen this Daniel Biss paper?

The book with those words was published in 2006, before the retraction of Biss' major results on combinatorial differential geometry. In the cited paper, Biss had published an amazing breakthrough ...
• 35.4k

### So what is Cohomology?

Some very basic answers, with the aim of giving you an idea of the big picture: On the most basic level, you can think of cohomology as a fancy way of counting/classifying holes in an underlying space ...
• 226k
Accepted

### Homology in a picture? (Is this picture just metaphorical, or a rigorous example that can be formalized?)

This is rigorous; the loop described by image $(1)$ is homologous to the "loop" described by image $(4)$ in the singular homology of $X:=\Bbb R^2\setminus\{0\}$ and we could convert the ...
• 41.3k
Accepted

### How does one orient a simplicial complex?

As Eric Wofsey says, there are two notions here: An orientation of an $n$-dimensional simplicial complex is a choice of orientation for each $n$-dimensional simplex. The easiest way to find one is to ...
• 61.9k

• 7,909

### Understanding the $\Delta$-complex structure of a quotient space

The identification diagrams are not quotients of a delta complex, but rather delta complex structures on the quotient space for the square itself. Delta complexes don't behave particularly well under ...
• 19.5k

### How does one orient a simplicial complex?

Let $S$ be the set of all the vertices of your simplicial complex. If you choose a total order $<$ on the set $S$, then $<$ is also a total order on any subset of $S$. In particular, the set ...
• 332k

### Let $K$ be a simplicial complex (that need not be finite). Prove $|K|$ is Hausdorff.

The space $|K|$ can be modelled in this way: $|K|$ is the set of maps $f$ from $V$ (the vertex set) to $\Bbb R_{\ge0}$ with the properties that $\{v\in V:f(v)>0\}$ is the vertex set of a simplex in ...
• 159k

### What meant by a $k$-chain in a simplicial complex?

Coefficients in a formal sum count the generators of the free abelian group. If $t_1,t_2, ...$ are 1-simplices, then $3t_1+4t_2+t_3$ is a collection of sticks: $3$ sticks at $t_1$, $4$ sticks at $t_2$...
• 1,523
Accepted

### Noncompact manifold as a simplicial complex

Infinite simplicial complexes are standard and useful things. I'm not sure what kinds of detail you want to know, but there's not too much to say, other than that finite simplicial complexes are ...
• 122k

### Understanding the $\operatorname{link}$ of a simplex $\sigma$

There is a nice picture of what links are on Wikipedia: The link of a simplex $\sigma$ is the set of simplices $\tau$ that don't intersect $\sigma$ but such that $\sigma \cup \tau$ is also a simplex. ...
• 54.4k
Accepted

### Homology groups with different complexes

To be precise, we have a way to define homology for each different kinds of cell complexes: On a simplicial complex one can define homology groups known as simplicial homology. On a $\Delta$-complex ...
• 332k
Accepted

### Necessary condition on homology group of a set to be contractible

The first homology group is far from enough to detect contractibility, since spaces can have non-vanishing higher homology groups. It's not even enough to have $H_n(X;G)$ vanish for every $n$ and $G$...
• 9,330
Accepted

### Why does the Betti number give the measure of k-dimensional holes?

This sentence in the paragraph just before the line from wikipedia you quote It follows that the homology group Hk(S) is nonzero exactly when there are k-cycles on S which are not boundaries. In ...
• 96.3k

### On simplicial complexes and their geometric realization

In the comments, where you rephrased your question, you expressed your concern regarding this statement: ... it could in principle happen that the intersection of two simplices is a union of several ...
• 122k
Accepted

• 122k

### When is simplicial complex a manifold?

Disclaimer: I know nothing of how to prove all this, I just know the results. This is the simplicial complex recognition problem. It can be phrased so as to make it algorithmic, because a simplicial ...
• 2,336
Accepted

### Triangulation of torus $\mathbb{S}^1\times \mathbb{S}^1$ with a minimal number of triangles

I don't think the answer above is a valid triangulation of the torus, for there would be two 1-dimensional simplices connecting the vertex of the square and the midpoint of each edge of the square. ...
• 434
Accepted

### Why is the first homology group of the torus is $\mathbb{Z}\oplus\mathbb{Z}$ instead of $\mathbb{Z}*\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$?

The notation $*$ denotes the free product of groups, which notably is an operation on groups, not just abelian groups. So $\mathbb{Z}*\mathbb{Z}$ is the free (nonabelian) group on two generators, not ...
• 332k
Accepted

### Extensions of Sperner families on finite sets

The best I can think of is to use the inclusion-exclusion principle to count the number of sets $\sigma\subset V$ which neither contains nor is contained in any of the $\sigma_i$ for $i=1,\ldots,m$. ...
• 9,747
Accepted

### Is there a triangulation of a closed surface with each vertex incident to $n\ge 7$ triangles?

There does always exist such a surface. There is a reasonably short proof using a big tool, namely Selberg's Lemma that every linear group has a torsion free subgroup of finite index. Assuming ...
• 122k
Accepted

### How is the sum of simplices defined?

The short answer is: the maps $\partial$ are defined on the free abelian groups generated by $n$-simplices, hence it makes sense to speak about sums of simplices in these groups. I'll try to give a ...
• 17.2k
Accepted

### Singular Homology is a special case of Simplicial Homology

Although the statements in your post are all true, none of them are correct interpretations of this passage from Hatcher's book. Instead, what Hatcher is saying is that for any topological space $X$ ...
• 122k