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Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.

The Shafarevich Conjecture states: $Gal(\bar Q / Q_{cycl})$ is free. That is, the Galois group of the algebraic closure of the rationals over the cyclotomic closure of the rationals is a free group.

References for Morava's thoughts are here and here and here and here.

This is exciting material, but I'm having trouble coming up with a way to summarize the gist and have some questions.

(1)What exactly is Morava's definition of a mixed Tate motive?

(2) What exactly is the connection Morava is advocating between number theory and geometric topology by invoking the appearance of the Riemann zeta function in Waldhausen's A-theory/pseudo-isotopy?

(3) Morava states that the map from the K-theory of the integers to that of the sphere spectrum, $K(\mathbb {Z}) \to K(\mathbb {S})$, is a rational equivalence as a (partial) explanation of (2). How exactly does this work??

(4) Where does Shafarevich fit in here?

Down-to-earth answers to these would be much appreciated!!

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Just a minor comment: "free profinite" is not the same as "free and profinite". – Mark Sapir Sep 23 '10 at 11:58
Thanks, edited to eliminate confusion. – Romeo Sep 23 '10 at 12:10
Actually, the conjecture is that the absolute Galois group in question is a "free profinite group". It is obviously profinite, but the point is that it is expected to be free in the category of profinite groups. That is different from being free in the category of all groups. See e.g. Neukirch, Schmidt and Wingberg: "Cohomology of Number Fields", page 449 or so. – John Rognes Nov 12 '10 at 15:29

(3) The statement is that the map of ring spectra $S \to HZ$ induces a rational equivalence $K(S) \to K(Z)$. A reference is Proposition 2.2 in:

Waldhausen, Friedhelm: Algebraic $K$-theory of topological spaces. I. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 35--60, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.

The proof uses the plus-construction definition of algebraic $K$-theory. The map $BGL(S) \to BGL(Z)$ is a $\pi_1$-isomorphism and a rational equivalence, since $\pi_{n+1} BGL(S) = \pi_n GL(S)$ is the group of infinite matrices over $\pi_n(S)$ for $n\ge1$, which is torsion. Hence $BGL(S)^+ \to BGL(Z)^+$ is also a rational equivalence.

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Ah, then (3) is relatively straightforward, thanks for the reference. Can you address how this related to (2)? This is still somewhat mysterious to me. – Romeo Oct 1 '10 at 0:37

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