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The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the theory of formal groups, local fields and moduli spaces. In another one there is a whole subject of derived algebraic geometry, as well as more esoteric objects like chiral homology and their applications. Perhaps this is wishful thinking, but I would really like to view these two fields as different facets of some deeper theory.

In particular, since the days of yore does AT employ a theory of localizations and completions at ideals of $\mathbb Z$. This already makes it feel as if we're really doing algebraic geometry over $\mathrm{Spec}\ \mathbb Z$. This feeling is supported by Nishida's theorem, which tells us that the sphere spectrum can be really thought about as some very nice nilpotent thickening of $\mathbb Z$. With a bit of white magic we can even apply local class field theory to the study of homotopy groups. However, there is a glaring issue: in number theory we really should be working not over $\mathbb Z$, but over some compactification of it, and should include information at infinite primes. Surprisingly, I have never seen infinite primes mentioned in homotopy theory.

An obvious guess that we should study localizations w.r.t. $H\mathbb R$ instead of $H\mathbb Q$ or $H\mathbb{F}_p$ fails for a simple reason that rational cohomology 'equal' real and complex ones. Trying to google for something like "infinite primes in algebraic topology" or "infinite prime Bousfield localization" returned zero relevant results. My knowledge about infinite primes in number theory is very limited, but as far as I understand constructions mostly revolve around completions with respect to archimedean norms on extensions of $\mathbb Q$ and studying real vector bundles with metric (well, it's mostly the same). I see no way to push either of these approaches to homotopy (metric cohomology? wut?).

Thus the question as stated: is there any theory exploiting some constructions (especially some form of localization) with respect to infinite primes in number fields to gain homotopy-theoretic results? Some variant of topological Arakelov geometry would be close to the ideal result, but I don't expect that it exists, so would welcome any leading threads. More broadly, what could take the role of those infinite primes and supply the missing (in what sense?) homotopy-theoretic information?

Of course, one could note that perhaps a more basic question is how the Galois extensions of $\mathbb Q$ can be generalized to spectra (because otherwise there are not many potential infinite primes around), but I don't feel like it should really be an obstruction. In any case it should be a subject of another question.

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  • $\begingroup$ I haven't heard of anything like this. I suppose in some sense one could try to think of "valuations" as corresponding to Bousfield localizations, or at least, localizations that behave like completion (e.g. localization at HF_p or Morava K-theory). There may be a fruitful analogy to be made here between certain Bousfield localizations and a notion of "valuations" on the sphere spectrum. I guess in some sense topology doesn't SEE the Archimedean place. $\endgroup$ – Jonathan Beardsley Oct 12 '14 at 0:02
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    $\begingroup$ I would say the people who could really speak to this are Andrew Salch and Jack Morava, neither of whom are, unfortunately, on MathOverflow. $\endgroup$ – Jonathan Beardsley Oct 12 '14 at 0:07
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    $\begingroup$ I remember reading in a problem session from a long-ago conference a question asked by Haynes Miller saying something like: "Lots of theorems work for all sufficiently large $p$ --- can we make a theory of the infinite prime?" $\endgroup$ – Jeff Strom Oct 12 '14 at 3:16
  • $\begingroup$ I'm not a topologist, so this may be too naive, but we know from work of Quillen and Sullivan that rational homotopy theory is equivalent to the homotopy theory of a DGL or DGA over $\mathbb{Q}$. We could simply tensor this by $\mathbb{R}$ couldn't we? I know that the "real" in the paper "Real homotopy theory of theory of Kahler manifolds" by Deligne, Griffiths, Morgan, Sullivan refers to this process. $\endgroup$ – Donu Arapura Oct 12 '14 at 7:26
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    $\begingroup$ I saw a talk by Morava in 2009, where he displayed a picture of the "Berkovich spectrum of the sphere spectrum". All of the finite prime branches had extended bits corresponding to homotopy-theoretic localizations. The archimedean branch ran into a picture of a dragon, labeled "$C^*$-algebras?". $\endgroup$ – S. Carnahan Oct 19 '14 at 3:09
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The Bousfield-Kan $p$-completion of a simplicial set $X_\bullet$ is the totalization (= homotopy limit) of the cosimplicial space obtained by levelwise iterating the functor $S\mapsto \mathbb F_p[S]$ (that sends a set $S$ to the free $\mathbb F_p$-vector space on that set).

I had an idea at some point (with the explicit thought that it could be something like Bousfield-Kan completion at the infinite place) of doing the same construction with the functor $S\mapsto B_1(\ell^1(S))$ in place of $\mathbb F_p[-]$. This makes sense because $B_1(\ell^1(-))$ is also a monad.

Here, $B_1(\ell^1(S))$ is the unit ball in the Banach space $\ell^1(S)$ (over the reals). A possible variant is to only use the positive part of $B_1(\ell^1(S))$.

I never pursued that idea. Tilman Bauer and I discussed it at some point, and we had the vague though that this might be related to the concept of $\ell^1$-homology. [Note that $\ell^1$-homology only sees $\pi_1$ and is completely insensitive to higher homotopy groups (and moreover, it vanishes identically when $\pi_1$ is ameanable), so it belongs more to the area of geometric group theory and less to algebraic topology.]

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  • $\begingroup$ André, I doubt that in this case that could be the infinite prime. All purely homotopy-theoretic information is already contained over $Spec \mathbb Z$, since we can reconstruct a homotopy type from its rationalization and completions. The infinite prime should contain some information that is inaccessible to the classical theory. For example, I have always wondered why smooth manifolds per se (not just their homotopy type) play such a big role in the study of homotopy. Or perhaps the infinite prime could hold some unstable information (but that is troublesome to interpret arithmetically). $\endgroup$ – Anton Fetisov Oct 18 '14 at 21:00
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    $\begingroup$ Your claim that we can reconstruct a homotopy type from its rationalization and completions only applies to nilpotent spaces. The kind of spaces seen by $\ell^1$-homology are very far from being nilpotent. $\endgroup$ – André Henriques Oct 18 '14 at 21:07
  • $\begingroup$ True, but the correct notion can't depend only on the fundamental group. I would not be surprised if differential cohomology would happen to be some invariant of "completed homotopy types". At least it can be restricted to the open subscheme, giving you merely a spectrum. But I can't see what would be then its fiber over $\infty$. "Just a differential form" doesn't look convincing. Are $l^1$ and differential cohomology related? $\endgroup$ – Anton Fetisov Oct 18 '14 at 21:32
  • $\begingroup$ I see, you seem to think that manifolds are things that have "completed homotopy types" (whatever that is)... that could be, but I wouldn't bet on it. What is the analog of a "completed homotopy type" in the world of pure algebra? Do you maybe want the homotopy groups of a "completed homotopy type" to be equipped with that kind of structure (whatever it is). PS: I am fairly certain that $\ell^1$ and differential cohomology are not related. $\endgroup$ – André Henriques Oct 18 '14 at 21:40
  • $\begingroup$ The settings are so different that it's hard to draw any direct analogies. However, I expect that homotopy theory over $\infty$ should be much simpler than over primes, it should be related to $\mathbb R$ with all of its metric and differential-geometric structure and also it should be actually something well-known, just not recognized as extra "completion" information (since I believe that in a field so venerable as AT all basic geometric ideas should be already discovered, and completion at primes should depend on something simple). I am not aware of any such structures on homotopy groups. $\endgroup$ – Anton Fetisov Oct 18 '14 at 22:05
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The only result in algebraic topology I know that explicitly involves both finite and infinite primes is Dustin Clausen's lift of Hilbert reciprocity to a statement about spectra. For each prime $p$ he introduces a $p$-adic version of the J-homomorphism and proves a product formula over all J-homomorphisms, including the usual one over $\mathbb{R}$, which reduces to Hilbert reciprocity after applying $\pi_2$.

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