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Can I find somewhere a table of the (first few) cohomology groups of $K(\mathbb{Z},n)$ with integer coefficients?

It seems like a natural counterpart to the table of the homotopy groups of spheres, but I couldn’t find anything. I’m aware of exposé 11, année 7 in the Cartan seminar where the homology of Eilenberg-MacLane spaces is computed, and I guess I could adapt it to the case where the group is $\mathbb{Z}$ and use the universal coefficient theorem to get the cohomology, but it’s not completely trivial, and I would be surprised that nobody thought about doing it before me.

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    $\begingroup$ I would guess that books about cohomology operations would be a good place to look $\endgroup$
    – Daniel Barter
    Commented Apr 27, 2016 at 16:20
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    $\begingroup$ The answer to the stable question (i.e. the integral cohomology of $H\mathbb{Z}$) is here: mathoverflow.net/questions/50519/…. Maybe there's something of interest for you there. $\endgroup$
    – Denis Nardin
    Commented Apr 27, 2016 at 16:20
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    $\begingroup$ I'll add that the lack of interest compared to the case of $F_p$ is perhaps due to the fact that $H\mathbb{Z}$ is not a flat cohomology theory, and so it has no nice Adams spectral sequence that could serve as an application of the structure of the integral Steenrod algebra. $\endgroup$
    – Denis Nardin
    Commented Apr 27, 2016 at 16:26
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    $\begingroup$ There is a paper "Integral cohomology operations" by Stan Kochman that treats the stable case, and this agrees with the unstable case through a range. However, the answer is unpleasant. I think that all possible applications can be done more cleanly using a combination of $H\mathbb{Z}/p$ and $H\mathbb{Q}$, or sometimes by using $K$-theory or complex cobordism instead. $\endgroup$
    – Neil Strickland
    Commented Apr 27, 2016 at 16:44

2 Answers 2

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As far as I know, there is no complete source besides the Cartan seminar. Of course, the homology computation gives the cohomology computation immediately. However, if you are only interested in the first few groups, then you might be able to get away with just iteratively using the Serre spectral sequences for the fibrations $$K(\mathbb{Z},n-1)\rightarrow\star\rightarrow K(\mathbb{Z},n).$$ Since you know the integral cohomology of $K(\mathbb{Z},2)$ with its ring structure, and since the spectral sequences play well with the ring structure, it is not a difficult exercise to compute $H^*(K(\mathbb{Z},n),\mathbb{Z})$ for $*\leq 3n$ or so for any given $n\geq 3$.

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There are several computations carried out explicitly in the paper

Samuel Eilenberg and Saunders Mac Lane, MR 65162 On the groups $H(\Pi,n)$. II. Methods of computation, Ann. of Math. (2) 60 (1954), 49--139.

For instance, I once wanted to know the groups $H_{4+i}(K(\mathbb{Z},4);\mathbb{Z})$ for $i\le 3$, and was able to extract the answer from Section 24.

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  • $\begingroup$ Was this by chance for a MU or MO-bordism computation? $\endgroup$
    – Benjamin Antieau
    Commented Apr 27, 2016 at 20:37
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    $\begingroup$ @BenjaminAntieau: Ha ha, almost! We were working out the first few stages of the Postnikov tower of $MO(2)$, in order to study realization of codimension 2 cohomology classes by embeddings. $\endgroup$
    – Mark Grant
    Commented Apr 27, 2016 at 20:49
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    $\begingroup$ @MarkGrant that is so flippin' cool. I will have to look up that paper. I guess it is obvious which it is from the title? $\endgroup$
    – Sean Tilson
    Commented Apr 28, 2016 at 11:43
  • $\begingroup$ @SeanTilson: Thanks! And yes, I think there's only one paper with that title! $\endgroup$
    – Mark Grant
    Commented Apr 29, 2016 at 10:53

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