Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested in...). As examples, Cornell and Rosen develop basically all of genus theory from cohomological point of view, a significant chunk of class field theory is encoded as a very elegant statement about a cup product in the Tate cohomology of the formation module, and Neukirch-Schmidt-Wingberg's fantastic tome "Cohomology of Number Fields" convincingly shows that cohomology is the principal beacon we have to shine light on prescribed-ramification Galois groups.

Of course, we also know that group cohomology can be studied via topological methods via the (topological) group's classifying space. My question is:

Question: Why doesn't this actually happen?

More elaborately: I'm fairly well-acquainted with the "Galois cohomology for number theory" literature, and not once have I come across an argument that passes to the classifying space to use a slick topological trick for a cohomological argument or computation (though I'd love to be enlightened). On the other hand, for example, are things like Tyler's answer to my question

Coboundary Representations for Trivial Cup Products

which strikes me as saying that there may be plenty of opportunities to carry over interesting constructions and/or lines of reasoning from the topological side to the number-theoretic one.

Maybe the classifying spaces for gigantic profinite groups are too hideous to think about? (Though there's plenty of interesting Galois cohomology going on for finite Galois groups...). Or maybe I'm just ignorant to the history, and that indeed the topological viewpoint guided the development of group cohomology and was so fantastically successful at setting up a good theory (definition of differentials, cup/Massey products, spectral sequences, etc.) that the setup and proofs could be recast entirely without reference to the original topological arguments?

(Edit: This apparently is indeed the case. In a comment, Richard Borcherds gives the link http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183537593 and JS Milne suggests MacLane 1978 (Origins of the cohomology of groups. Enseign. Math. (2) 24 (1978), no. 1-2, 1--29. MR0497280)., both of which look like good reads.)

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    $\begingroup$ My impression is that many automorphic forms describe cohomology classes on arithmetic manifolds or orbifolds, which are classifying spaces for their fundamental groups. Also, the class number of a quadratic imaginary number field is the number of cusps of the corresponding Bianchi orbifold, which could be turned into a cohomological statement. But I'm not sure this is related to your question. $\endgroup$ – Ian Agol Aug 31 '10 at 4:32
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    $\begingroup$ You might be slightly interested in the answer by Josh Roberts in this thread: mathoverflow.net/questions/10879/intuition-for-group-cohomology $\endgroup$ – David Corwin Aug 31 '10 at 4:36
  • $\begingroup$ @Agol: Well, perhaps not directly, but it sounds fascinating regardless, so +1. If there's an answer there to be elaborated on, I'd love to see it. This will save me from asking the new question "What was Agol talking about when he said..." :) $\endgroup$ – Cam McLeman Aug 31 '10 at 12:29
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    $\begingroup$ Related: Trees and more general buildings can give you the total spaces of the universal bundle for the group they were constructed from. In any case their homology is a G-module. And buildings are used in arithmetic, not directly for Galois groups though, as far as I know, you rather look e.g. at SL_n(L) and then get an action of Gal(L/k) on the result... $\endgroup$ – Peter Arndt Sep 1 '10 at 13:09

Classifying spaces are widely used in algebraic number theory, but in slightly disguised form. A classifying space is really just an approximation to the classifying topos of a group. However the classifying topos is just the category of G-sets, which is exactly what one uses in defining group cohomology and so on. Or to put it another way, all the useful information in the classifying space is already contained in the category of G-sets.

The comment at the end of the question is correct: group cohomology was discovered as the cohomology of the classifying space, and the topological constructions were then turned into algebraic constructions and removed from the theory. So in some sense all the group cohomology calculations are implicitly using the classifying space.

  • $\begingroup$ +1: Thanks for this interpretation -- I think this makes a lot of sense, though I'm going to spend some time working it out more carefully. Do you have a reference for the historical comment? I'd like to understand the historical development better. $\endgroup$ – Cam McLeman Aug 31 '10 at 12:43
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    $\begingroup$ From my Class Field Theory Notes p86. In the mid-1930s, Hurewicz showed that the homology groups of an "aspherical space" $X$ depend only on the fundamental group $\pi$ of the space. Thus one could think of the homology groups $H_{r}(X,\mathbb{Z})$ of the space as being the homology groups $H_{r}(\pi,\mathbb{Z})$ of the group $\pi$. It was only in the mid-1940s that Hopf, Eckmann, Eilenberg, MacLane, Freudenthal and others gave purely algebraic definitions of the homology and cohomology groups of a group $G$. $\endgroup$ – JS Milne Aug 31 '10 at 14:27
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    $\begingroup$ Continued: It was then found that $H^{1}$ coincided with the group of crossed homomorphisms modulo principal crossed homomorphisms, and $H^{2}$ with the group of equivalence classes of "factor sets", which had been introduced much earlier (e.g., I. Schur, \"{U}ber die Darstellung der endlichen$\ldots$ , 1904; O. Schreier, \"{U}ber die Erweiterungen von Gruppen, 1926; R. Brauer, \"{U}ber Zusammenh\"{a}nge$\ldots$ , 1926). For more on the history, see MacLane 1978 (Origins of the cohomology of groups. Enseign. Math. (2) 24 (1978), no. 1-2, 1--29. MR0497280). $\endgroup$ – JS Milne Aug 31 '10 at 14:29
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    $\begingroup$ projecteuclid.org/euclid.bams/1183537593 $\endgroup$ – Richard Borcherds Aug 31 '10 at 14:32

As a topologist, my view is that group cohomology of interest to number theorests seems to generally be with non-trivial module coefficients. Many of the tricks topologists employ to study spaces do not apply in this setting, and indeed there are some topologists whose view is that once the coefficients are non-trivial then one is "just doing algebra." But attitudes can change: cohomology of profinite groups was perhaps viewed similarly, but in the "chromatic" (number theoretic) framework for stable homotopy, cohomology of profinite groups figures prominently. For this reason, some topologists work hard to show that techniques which work for finite groups pass sensibly to the profinite settings.

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    $\begingroup$ One interesting special case: if I remember correctly, the $E^2$-term of the Adams-Novikov spectral sequence is isomorphic to the continuous cohomology of the Morava stabilizer group (the automorphism group of the Honda formal group law) with coefficients the Lubin-Tate universal deformation ring. For the relevance of Lubin-Tate theory in class field theory, this might be called a number-theoretic object. But the usual methods for computation are of a more topological kind. See math.rochester.edu/people/faculty/doug/mu.html#repub , especially chapter 7. $\endgroup$ – Lennart Meier Aug 31 '10 at 11:33
  • $\begingroup$ This answer agrees very much with my impression: it's not so much finite or pro-finite that's the issue, but the focus on trivial or non-trivial coefficients. I do wish topologists would think more about non-trivial coefficients, since they're so good at coming up with ideas useful for arithmeticians. $\endgroup$ – Minhyong Kim Aug 31 '10 at 12:06
  • $\begingroup$ @Dev Sinha: Okay, fair point. On the other hand, there are a lot of important statements in group cohomology which involve trivial coefficients: The Schur multiplier (or Hopf's Integral Homology Formula) comes to mind, as does the interpretation of the relation rank of a pro-$p$-group $G$ as $\operatorname{dim}H^2(G,\mathbb{F}_p)$. $\endgroup$ – Cam McLeman Aug 31 '10 at 12:25
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    $\begingroup$ I must admit, I twisted something in my head. The cohomology of the Morava stabilizer group is the $E^2$ term of a spectral sequence computing the stable homotopy groups of the $K(n)$-localized sphere, while in the book by Ravenel the Adams-Novikov $E^2$-term for the ($p$-localized) sphere is computed. But I think, my point remains probably valid. $\endgroup$ – Lennart Meier Aug 31 '10 at 12:37
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    $\begingroup$ just another example like Lennart's, specifically an example with non-trivial coefficients and number theoretic objects, there is the homotopy fixed point spectral sequence which has $E^2=H^*(\mathbb{G};\pi_*(E_n))$ converging to $\pi_*(E^{h\mathbb{G}}_n)=\pi_*(EO_n)$, here G is some (possibly extended) morava stabilizer group which acts on $E_n$ (lubin-tate theory, $\pi_*(E_n)$ is the ring that classifies universal deformations of the Honda height n formal group law) and $EO_n$ is called higher real k-theory. ... $\endgroup$ – Sean Tilson Sep 1 '10 at 13:01

I suppose we should also mention algebraic k-theory. Quillen defined the k-groups as the homotopy groups of certain classifying spaces. For a unital, associative ring $R$, $$ K_n(R):=\pi_n(BGL(R)^+), $$ where $GL(R)$ is the direct limit of the general linear groups and $^+$ is Quillen's plus-construction on spaces whose fundamental groups have perfect subgroups. Now I'm not sure how useful this has been for computation (these are homotopy groups, after all), but the classifying space is used. And there are number theory applications of algebraic k-theory.


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