Question

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed links. How would that be with Spec Z? Then, topologists have things like virtual 3-manifolds, has that analogies in arithmetics?

Edit: "What is the analogy of quantum invariants in arithmetic topology?", "If a prime number is a knot, what is a crossing?" asks this old report.

An other such question: Minhyong Kim stresses the special complexity of number theory: "To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck." Probably a feeling of such a special complexity makes one initially interested in NT. Knot theory is an other case inducing a similar impression. Could both cases be connected by the analogy above? How could a precise description of such special complexity look like and would it cover both cases? Taking that analogy, I'm inclined to answer Minhyong's question with the contrast between low-dimensional (= messy) and high-dimensional (= harmonized) geometry. Then I wonder, if "harmonizing by increasing dimensions"-analogies in number theory or the Langlands program exist.

Minhyong hints in a mail to "the study of moduli spaces of bundles over rings of integers and over three manifolds as possible common ground between the two situations". A google search produces an old article by Rapoport "Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen" (Analogies between moduli spaces of vector bundles and flags) (p. 24 here, MR). There, Rapoport describes the cohomology of such analogous moduli spaces, inspired by a similarity of vector bundles on Riemann surfaces and filtered isocrystals from p-adic cohomologies, "beautifull areas of mathematics connected by entirely mysterious analogies". (book project by R., Orlik, Dat) As interesting as that sounds, I wonder if google's hint relates to the initial theme. What do you think about it? (And has the mystery Rapoport describes now been elucidated?)

This interesting essay by Gromov discusses the topic of "interestung structures" in a very general way. Acc. to him, "interesting structures" exist never in isolation, but only as "examples of structurally organized classes of structured objects", Z only because of e.g. algebraic integers as "surrounding" similar structures. That would fit to the guesses above, but not why numbers were perceived as esp. fascinating as early as greek antiquity, when the "surrounding structures" Gromov mentions were unknown. Perhaps Mochizuki has with his "inter-universal geometry" a kind of substitute in mind?

Answer 1

The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields" (arXiv:0107210): the author states the Poincare conjecture as "S³ is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.

The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.

Answer 2

I think it's important to keep track of the fact that the analogy isn't between individual number fields and individual 3-manifolds; it's between the collection of all number fields and the collection of all 3-manifolds. So in my opinion it's slightly awry to ask for an "arithmetic Poincare conjecture" about Spec Z; I don't think Spec Z should be thought of as analogous to S^3 in any meaningful sense.

As always, John Baez has useful things to say.

I saw Deninger give a beautiful talk about his point of view on this, some of which is recorded in this paper. Part of the idea, somewhat vaguely, is that you should think of a number field not as an unadorned 3-manifold but as a 3-manifold with a flow on it. And then the finite primes are not just knots, but closed orbits of that flow! That gives a more satisfying answer to "why should a 3-manifold have a distinguished countably infinite family of knots on it," makes the connection with dynamical zeta functions, etc.

Answer 3

This has been well-addressed by the answerers before me, but just to chime in -- there are a variety of analogs one could make for the Poincare conjecture for number fields. For one, there are several equivalent statements about the Poincare conjecture for 3-manifolds which are not equivalent when transferred over by analogy to the number field case. As a first easy example, while 3-manfiolds enjoy a clean Poincare duality, number fields have extra 2-torsion. In particular, one frequently has $H^1(\mathcal{O}_K,\mathbf{G}_m)$ trivial with $H^2(\mathcal{O}_K,\mathbf{G}_m)$ non-trivial (example: any real quadratic number field with trivial class group). The equivalences (or lack thereof) between being an integral homology 3-sphere, a rational homology 3-sphere, and a homotopy 3-sphere are not the same in the two "categories." So depending on how you phrase your analogous Poincare conjecture, you may get different answers. The cleanest form (found in Niranjan Ramachandran's "A Note on Arithmetic Topology", which deals exclusively with this question) is that there are exactly ten rational homology 3-spheres which are homotopy 3-spheres, namely the 9 quadratic imaginary number fields of class number one and $\mathbb{Q}$ itself. (Or really, $\mathbb{Z}$ itself), and even more homotopy 3-spheres.

A second frequently under-emphasized point to make is that no one really knows what the right category for this analogy is on the number theory side. As mentioned above, if you take your category to be Specs of rings of integers in a number field, you don't get the Poincare conjecture. On the other, if you take the point of view of Artin-Verdier theory (or alternatively, Arakelov theory), where you include in your spaces some information about the behavior of the infinite primes (from the point of view of number theory, defining Spec(Z) as the set of prime ideals ignores the obviously important primes at infinity), then you get a different cohomology theory. With these new cohomology groups in place, some things look a little bit cleaner. Again, see Ramachandran.

Answer 4

I cannot really say anything about relations with Poincare conjecture, but the obvious references you should look at are

M. Morishita. On certain analogies between knots and primes. Journ. f¨ur die reine u. angew. Math., 550 (2002), 141–167. (there is a more recent exposition also by Morishita in http://arxiv.org/abs/0904.3399 ) and Manin's "The notion of dimension in geometry and algebra": http://arxiv.org/abs/math/0502016 which contains abundant references inside.

Answer 5

There are some cryptic remarks about this in the first few pages of this talk of Fujiwara:

http://www.ms.u-tokyo.ac.jp/~t-saito/conf/rv/Leopoldt.pdf

(n.b. I believe - though I'm not 100% sure - that some of the later material in these slides has been retracted. But the relevant part is early on.)

Answer 6

From reading the Morishita article 0904.3399 (page 24), there is a following analogue of Poincare conjecture:

Suppose that k is a number field whose ring of integers is “cohomologically ”, namely for i ≥ 0. Then must be .