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63 votes
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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 ...
zyx's user avatar
  • 35.4k
24 votes

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 ...
Ben Grossmann's user avatar
19 votes
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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 ...
FShrike's user avatar
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13 votes
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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 ...
David E Speyer's user avatar
9 votes

Surjection from cohomotopy to cohomology

Identify $$\pi^nX=[X,S^n]\quad \text{and}\quad H^nX=[X,K(\mathbb{Z},n)].$$ The comparison map $$\Phi:\pi^nX\rightarrow H^nX$$ is induced by a choice of homotopy class of map $\varphi:S^n\rightarrow K(\...
Tyrone's user avatar
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8 votes

Why is this not a triangulation of the torus?

Better late than never. The definition of a simplicial complex reads as follows. A finite (Euclidean) simplicial complex is a finite collection $K$ of simplices in some Euclidean space $\mathbb{R}...
TheGeekGreek's user avatar
  • 7,909
7 votes

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 ...
Kyle Miller's user avatar
  • 19.5k
7 votes

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 ...
Eric Wofsey's user avatar
7 votes

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 ...
Angina Seng's user avatar
7 votes

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 $t_2$...
autodavid's user avatar
  • 1,523
6 votes
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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 ...
Lee Mosher's user avatar
  • 122k
6 votes

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. ...
Najib Idrissi's user avatar
6 votes
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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 ...
Eric Wofsey's user avatar
6 votes
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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 $G$...
William's user avatar
  • 9,330
6 votes
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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 ...
Ethan Bolker's user avatar
  • 96.3k
6 votes

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 ...
Lee Mosher's user avatar
  • 122k
6 votes
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link of a vertex in the triangulation of $S^k$ is a triangulation of $S^{k-1}$?

The link is a homology sphere, ie has the homology of a sphere. To see this, denote the link at $v$ by $L$ and the star at $v$ by $S$. Note that $L$ is homotopy equivalent to $S - v$ and both $S$ and $...
ronno's user avatar
  • 12k
6 votes

When is simplicial complex a manifold?

One very important result of geometric topology is the double suspension theorem of Bob Edwards, in which he started with a certain triangulated 3-manifold $M$ having the same homology groups as the $...
Lee Mosher's user avatar
  • 122k
6 votes

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 ...
Thomas Anton's user avatar
  • 2,336
5 votes
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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. ...
Greywhite's user avatar
  • 434
5 votes
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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 ...
Eric Wofsey's user avatar
5 votes
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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 $i=1,\ldots,m$. ...
Einar Rødland's user avatar
5 votes
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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 ...
Lee Mosher's user avatar
  • 122k
5 votes
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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 ...
qualcuno's user avatar
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5 votes
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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 $X$ ...
Lee Mosher's user avatar
  • 122k
4 votes

Has anyone ever actually seen this Daniel Biss paper?

I think the key word here is "apparently", and the full quote from Green includes a confession that he doesn't really know what he's talking about: "I talk about [math] a lot, and I think about it ...
Jason's user avatar
  • 15.4k
4 votes

k-connectedness of simplicial complexes

No, there is no such algorithm. Here's why. Given a finite simplicial complex $\mathcal{C}$ and a choice of vertex $v$, there is an algorithm to write down a finite presentation of $\pi_1(\mathcal{C},...
Lee Mosher's user avatar
  • 122k
4 votes
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Product of standard simplexes is homeomorphic to a standard simplex

Let's try and see if your proof works for $n = m = 1$. Write $$ \Delta_1 = \{ (t, 1 - t) \, | \, t \in [0,1] \} $$ and use the $s$-parameter for the second copy of $\Delta_1$. Then you suggest ...
levap's user avatar
  • 65.8k
4 votes
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What are some important (discrete) theorems involving (abstract) simplicial complexes

There are many theorems "devoted to (abstract) simplicial complexes". Here are a few examples: The Kruskal–Katona theorem, characterising $f$-vectors of simplicial complexes. The Alexander duality ...
Dietrich Burde's user avatar
4 votes

Extensions of Sperner families on finite sets

If your concern is computational complexity then one can avoid the term $2^m$ which tends to be doubly exponential in $n$. One can solve this problem in time $\text{O}(mn 2^n)$ simply by checking for ...
Tony Blair's Witch Project's user avatar

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