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A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...

5 votes

How to get product on cohomology using the K(G, n)?

A very specific example: for $G=\mathbb{Z}$ and $n=m=1$, it is the map $S^1 \times S^1 \to CP^\infty$ which maps $S^1 \times S^1$ to the 2-skeleton $S^2$ of $CP^\infty$ by a degree 1 map. Here's a gen …
Aaron Bergman's user avatar
58 votes

Cohomology and fundamental classes

This is a reply to Alon's comment, but it's too long to be a comment and is probably interesting enough to be an answer. Here's an example Thom gives of a homology class that is not realized by a subm …
Eric Wofsey's user avatar
  • 31.2k
14 votes
1 answer
892 views

Commutativity in K-theory and cohomology

However, it is not an equivalence over $\mathbb{Z}$ because the cohomology of $BU$ is just a polynomial algebra and has no Steenrod operations. … Steenrod operations can be understood as obstructions to the cup product on ordinary cohomology being strictly commutative. …
6 votes

Cohomology and Eilenberg-MacLane spaces

Here's a precise statement: reduced singular cohomology $H^n(X;G)$ is naturally isomorphic to homotopy classes of pointed maps from $X$ to $K(G,n)$, for any pointed topological space $X$ having the homotopy … In particular, to do de Rham cohomology you presumably want $G$ to be $\mathbb{R}$ or $\mathbb{C}$, and then $K(G,n)$ is really monstrous geometrically. You may want to take a look at this question. …
Community's user avatar
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11 votes
Accepted

What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

All cohomology in this answer will have $\mathbb{Z}/2$ coefficients, and $K_n$ will denote $B^n(\mathbb{Z}/2)$. … By the Yoneda lemma, the cohomology $H^m(K_n)$ can also be thought of as the natural operations that take a cohomology class in degree $n$ on a space and give a cohomology class in degree $m$. …
Eric Wofsey's user avatar
  • 31.2k
16 votes
Accepted

Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

Let $H\mathbb{Z}$ denote the spectrum for cohomology with coefficients in $\mathbb{Z}$, so your spectrum is $H\mathbb{Z}\wedge M_k(G)$. … Thus up to a degree shift, the cohomology theory $H\mathbb{Z}\wedge M_k(G)$ satisfies the Eilenberg-Steenrod dimension axiom, so it must coincide with $H^{*+k}(X,G)$. …
Eric Wofsey's user avatar
  • 31.2k
7 votes

What is a cup-product in group cohomology, and how does it relate to other branches of mathe...

The group cohomology $H^n(G;A)$ can be identified with the cohomology $H^n(BG;A)$ of the classifying space $BG$ of $G$ with coefficients in A (interpreting the action of $G$, which is the fundamental group … The cup product on group cohomology is then the same as the usual cup product of singular cohomology. …
David White's user avatar
  • 30.3k
8 votes
Accepted

Compactly supported cohomology of homotopy equivalent manifolds

Let $M$ be a punctured torus and $N$ be a twice-punctured plane. Then $M$ and $N$ are homotopy equivalent, but their one-point compactifications are not (the first being a torus and the second having …
Eric Wofsey's user avatar
  • 31.2k
67 votes

Can anyone give me a good example of two interestingly different ordinary cohomology theories?

cohomology. … What is of interest is when you remove the dimension axiom to get "extraordinary" cohomology theories, which Oscar talks about in his answer. …
Eric Wofsey's user avatar
  • 31.2k
10 votes
Accepted

"Skew Cohomology" of a Space

I believe it should be exactly the same as ordinary singular cohomology. … It should define a cohomology theory for the exact same reason that usual singular cohomology does (the usual proof of excision by subdivision seems to work since the cosubdivision of a $\Sigma_n$-invariant …
Eric Wofsey's user avatar
  • 31.2k
6 votes

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

These are related to singular cohomology by looking only at constant sheaves. … It follows, for example, that for any CW-complex or any manifold, singular cohomology agrees with both Cech and derived cohomology of constant sheaves. …
Eric Wofsey's user avatar
  • 31.2k
20 votes

What is (co)homology, and how does a beginner gain intuition about it?

Cup products give cohomology a natural graded ring structure, and the fact that this structure is preserved by continuous maps makes it often much easier to compute cohomology than homology. … Another (not unrelated) reason that cohomology can be easier to work with is that cohomology is a representable functor: H^n(X;A) is homotopy classes of maps from X to the Eilenberg-MacLane space K(A,n …
Eric Wofsey's user avatar
  • 31.2k