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47 votes
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Why is (co)homology useful and in which way?

This question is extremely broad, and a lot of ink has been spilled on the subject of "why is cohomology useful" (see, for example, here, here, here, here, here, here, here, or here, all ...
Chris Grossack's user avatar
28 votes

What is the solution to Nash's problem presented in "A Beautiful Mind"?

The problem is to find a subset $X$ of $\mathbb{R}^3$ such that if $V$ is the vector space of vector fields $F$ on $\mathbb{R}^3$\ $X$ with $\nabla\times F = 0$ and $W$ is the vector space of vector ...
Enrico M.'s user avatar
  • 26.2k
28 votes
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Vanishing differential forms in cohomology

No, this is not always possible. One can use differential forms to define higher order cohomology operations called Massey products and if they don't vanish, then you have an obstruction for the ...
levap's user avatar
  • 65.9k
26 votes

Meaning of "holes" counted by homology groups

The "dimension" of a hole is the dimension of the part that actually exists. For a sphere, then, we have a $2$-dimensional boundary, which is missing a $3$-dimensional ball inside it. We ...
Chris Grossack's user avatar
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
22 votes
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The First Homology Group is the Abelianization of the Fundamental Group.

I do not understand a lot of the last paragraph either, but I will give a slightly different proof of the statement based on Bredon, Topology and Geometry, p. 174. For every point $x\in X$, fix a ...
Benedikt Hunger's user avatar
22 votes
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Finding a mistake using Mayer-Vietoris

By your argument at the beginning, removing four points from a two-sphere should yield something which is homotopy equivalent to a zero-sphere. This is obviously not true. Your error is that after ...
Carsten S's user avatar
  • 8,755
20 votes
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What is the difference between cellular, simplicial and singular homology and their simplices?

Let's zoom out a bit. A definition of homology has to navigate a tradeoff between several different nice properties it could satisfy, most notably a tradeoff between how easy it is to compute in ...
Qiaochu Yuan's user avatar
19 votes
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Local Degree of a map between n-spheres

To make sense of a proof of this nature, it isn't enough to know that there exists a map from $H_d(V, V \backslash \{ y \})$ to $H_d(S^d, S^d \backslash \{ y \})$, and that there exists an isomorphism ...
Kenny Wong's user avatar
  • 32.3k
18 votes

Is there a homology theory that counts connected components of a space?

There is no homology theory which satisfies the following conditions: $H_0(X)$ is the free abelian group generated by the connected components of $X$. The homomorphism $f_*:H_0(X)\to H_0(Y)$ induced ...
Jon Barmak's user avatar
18 votes
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Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

No such groups exist. Suppose $\operatorname{Ext}(A,\mathbb{Z})=\operatorname{Hom}(A,\mathbb{Z})=0$. First, note that the functor $\operatorname{Ext}(-,\mathbb{Z})$ turns injections into surjections ...
Eric Wofsey's user avatar
18 votes
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Detail in the proof that sheaf cohomology = singular cohomology

To elaborate on Dmitri Pavlov's comment, your questions have been answered in a recent paper by Yehonatan Sella. Sella give an example (Example 0.3) of a locally contractible space $X$ for which $\...
Eric Wofsey's user avatar
17 votes

Surprising applications of cohomology

Close to a real-world application is maybe the application to mixed finite elements. Finite element exterior calculus In a nutshell: Instead of solving numerically $$\Delta u =0,$$ one approximates ...
Steffen Plunder's user avatar
16 votes

The First Homology Group is the Abelianization of the Fundamental Group.

I follow the proof by Hatcher that the OP outlines. The first important point is the following, once you have proved that $f = \Sigma_{i,j}(-1)^jn_i\tau_{i,j}$ for your 1-cycle $f$, you need to ...
A. Bordg's user avatar
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15 votes
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Homology and cohomology of 7-manifold

For any connected manifold $M$, there is a homomorphism $\pi_1(M)\to\mathbb{Z}/2$ which sends a loop to $0$ if going around the loop preserves orientation and sends the loop to $1$ if going around the ...
Eric Wofsey's user avatar
14 votes

Why are we interested in cohomology?

A major issue is the multiplicative structure that is around in cohomology. This allows you to distinguish spaces, which have the same homology. As an example, the $X:=\mathbb CP^2$ and $Y:=S^2\vee S^...
Andreas Cap's user avatar
14 votes
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induced homomorphism in homology

In terms of the logical process of inducing maps, you can think in the following way: A continuous map $f : X\rightarrow Y$ induces chain maps $f_\# : C_k(X) \rightarrow C_k(Y)$ (for each $k$) which ...
john's user avatar
  • 2,406
13 votes
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Direct proof that the wedge product preserves integral cohomology classes?

$\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}$I'm not sure if this is an answer to the question, since it does refer to $H_{\ast}(M, \ZZ)$, but I think it sheds some interesting light on why the problem is ...
David E Speyer's user avatar
13 votes
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Intuitive reason why the Euler characteristic is an alternating sum?

I don't know how intuitive this will be, but here is how I think of it. It is similar to what Travis mentioned in the comments. We need the Euler characteristic to not change when we change the ...
N. Owad's user avatar
  • 6,832
13 votes
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Why group cohomology and not group homology?

If you read, say, Brown's book, it becomes quite clear that cohomology is better than homology. Firstly, as Don Alejo noted, cohomology comes equipped with the cup-product, but also in regard to ...
Moishe Kohan's user avatar
13 votes

Nilpotent elements of cohomology ring

The Cech cohomology ring of a space $X$ is the direct limit of the cohomology rings of nerves of open covers of $X$. If $X$ is compact, then finite open covers are cofinal in all open covers, so ...
Eric Wofsey's user avatar
13 votes
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Computing the Simplicial Homology of $S^1$

Let us consider $S^1$ as a triangle, with vertices $u$, $v$, and $w$ and edges between each pair of them which we will write as $uv$, $vw$, and $uw$. We use the ordering $u<v<w$ on the vertices....
Eric Wofsey's user avatar
13 votes
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 ...
David E Speyer's user avatar
13 votes

Cohomology ring of a wedge sum

The right way to think of $H^*(X\vee Y)$ is not as a quotient of $H^*(X)\times H^*(Y)$ (which, as you point out, does not have a natural ring structure) but rather as a subring. Indeed, this is what ...
Eric Wofsey's user avatar
13 votes

Interpretation of Borel equivariant cohomology.

Borel cohomology $H^*_G(X)$ is the cohomology of the homotopy quotient $X//G := EG \times_G X$. The usual quotient, $X/G$, is bad from a homotopy theoretical point of view: it does not preserve ...
Najib Idrissi's user avatar
13 votes
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If a map induces the identity on homology, does it also on cohomology?

Let $X=S^3\vee\Sigma\mathbb{RP}^2$ and consider the following map $f:X\to X$. On the $S^3$ summand, $f$ is the identity. On the $\Sigma\mathbb{RP}^2$ summand, $f$ is the composition of the pinch map ...
Eric Wofsey's user avatar

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