TITLE: Questions about analogy between Spec Z and 3-manifolds
QUESTION [52 upvotes]: 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: New MFO report: "At the moment the topic of most active interaction between topologists and number theorists are quantum invariants of 3-manifolds and their asymptotics. This year’s meeting showed significant progress in the field."
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 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?)
Edit:
Lectures by Atiyah discussing the above analogies and induced questions of "quantum Weil conj.s" etc.
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?    
Edit: Hidekazu Furusho: "Lots of analogies between algebraic number theory and 3-dimensional topology are suggested in arithmetic topology, however, as far as we know, no direct relationship seems to be known. Our attempt of this and subsequent papers is to give a direct one particularly between Galois groups and knots."

REPLY [6 votes]: It seems the following remarks in M. Kapranov's paper http://arxiv.org/abs/alg-geom/9604018
page 64 bottom, has not been mentioned so far

According to the point of view going
  back to Y.I. Manin and B. Mazur, one
  should visualize any 1-dimensional
  arithmetic scheme X as a kind of
  3-manifold and closed points x ∈ X as
  oriented circles in this 3-manifold.
  Thus the Frobenius element (which is
  only a conjugacy class in the
  fundamental group) is visualized as
  the monodromy around the circle
  (which, as an element of the
  fundamental group, is also deﬁned only
  up to conjugacy since no base point is
  chosen on the circle), Legendre
  symbols as linking numbers and so on.
  From this point of view, it is natural
  to think of the operators (algebra
  elements) af,x,d for ﬁxed f and
  varying x, d as forming a free boson
  ﬁeld Af on the “3-manifold” X; more
  precisely, for ±d > 0, the operator a
  ± f,x,d is the dth Fourier component
  of Af along the “circle” Spec(Fq(x)).
  The bosons a ± f,x,d and their sums
  over x ∈ X (i.e., the Taylor
  components of log Φ ± f (t)) will be
  used in a subsequent paper to
  construct representations of U in the
  spirit of [FJ].

It might be that recent paper by Kapranov and coauthors: 
http://arxiv.org/abs/1202.4073
The spherical Hall algebra of Spec(Z)
is somehow developing ideas quoted above.
The question which I heard from V. Golyshev and others many years ago is the following:
if Spec (Z) is analogous to 3-fold, what should be arithmetic analog of Chern-Simons theory ?