# Group cohomology: cup product for cyclic groups

I have written almost everything and then I figured out where my (idiotic) problem has been... So I will post this question anyway, just so other people who are looking for this computation online can find an answer. Anything that is written here can be found at Cartan&Eilenberg.

My goal is to compute (as explicitly as possible) the cup product structure in Tate's cohomology of a cyclic group $G$.

I have searched a lot for an answer online and in many books, and the only place I have found what I was looking for is in Cartan&Eilenberg- Homological Algebra. When I tried to verify the equations given I ran across a problem, which was due to some silly mistake by me. The solution there goes something like this:

Let $X=X_\bullet$ be a complete resolution for $G$. By $X\otimes X$ we mean the double complex $X_r\otimes X_s$ with differentials $d'=d\otimes \text{id}$ and $d''=\text{id}\otimes d$. Then a mapping $\phi:X\to X\otimes X$ is defined to be a family of $G$-homomorphisms $\phi_{r,s}:X_{r+s}\to X_r\otimes X_s$ satisfying

1. $\phi_{r,s}d=d'\phi_{r+1,s}+(-1)^rd''\phi_{r,s+1}$.
2. $\epsilon=(\epsilon\otimes\epsilon)\phi_{0,0}$

where $\epsilon$ is the augmentation map $X_0\to\mathbb{Z}$ given in the complete resolution. (In the book they did not add the sign $(-1)^r$ but they have made some comment regarding the sign which I did not understand. The formula written here is correct).

From such a mapping we get a cup product defined on cochains by $f\smile g:=(f\otimes g)\phi_{r,s}$. My question is about the explicit construction of $\phi$ in the cyclic case.

So assume that $G$ is cyclic of order $n$ with generator $x$. We give a complete resolution by $X_i=\mathbb{Z}[G]$ and differentials $d_{2i}=N\cdot:X_{2i}\to X_{2i-1}$ (multiplication by $N=\sum x^k$), $d_{2i+1}=T\cdot:X_{2i+1}\to X_{2i}$ (multiplication by $T=x-1$), and the augmentation map $\epsilon:x_0\to \mathbb{Z}[G]$ by $\epsilon(1)=1$.

Now the definition of $\phi_{r,s}$ is given in the book by $$\phi_{r,s}(1) := \begin{cases} 1\otimes 1 & \text{if } r \equiv 0 (\text{mod }2)\\ 1\otimes x & \text{if } r+1\equiv s \equiv 0 (\text{mod }2)\\ \sum_{0\le l< k\le n-1}x^l\otimes x^k & \text{if } r\equiv s \equiv 1 (\text{mod }2) \end{cases}$$

My problem was that It seemd to me that the first property of a mapping does not hold. my mistake was that during my verification of the first property I did not use the correct dimensions for $d',d''$.

I'm currently writing this in more detail in sharelatex.com, I'll send a link here when I'll be done. (This will include the actual equations on the cohomology groups)

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Is it okay that I have posted a solved question? As I have written in the end, I will write about it in more details at sharelatex.com. should I add the link as an answer? – edo arad Jun 23 '13 at 9:23
I think it's fine though perhaps you could have asked first the question you had in mind and then you could have answered your own question... – DonAntonio Jun 23 '13 at 9:27
@DonAntonio thanks. done. – edo arad Jun 23 '13 at 10:12
@edoarad What Don meant was that you should officialy answet the question, that is, type the answer in the box below. That way this question won't come up as unanswered. We don't use the [SOLVED] system here. – Git Gud Jun 23 '13 at 10:38
@GitGud oh, that make sense.. – edo arad Jun 23 '13 at 11:20

## 1 Answer

I have written an assignment on group cohomology, which converges to describing the cup product structure on Tate cohomology of (finite) cyclic groups. There is also a very detailed section about Tate cohomology and a short section about the cup product.

https://www.sharelatex.com/project/51ad8e7a0053f29d3211bd1a

(hopefully this will be useful for someone but me..)

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