Why aren't there more classifying spaces in number theory? - MathOverflow

Why aren't there more classifying spaces in number theory?

2010-08-31T04:11:09Z

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

http://mathoverflow.net/questions/15375/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.)

Answer by Dev Sinha for Why aren't there more classifying spaces in number theory?

2010-08-31T05:39:23Z

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.

Answer by Richard Borcherds for Why aren't there more classifying spaces in number theory?

2010-08-31T12:41:00Z

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.

Answer by Josh Roberts for Why aren't there more classifying spaces in number theory?

2010-08-31T17:38:05Z

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.