$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: $\operatorname{Gal}(\bar \Q \,/ \,\Q_{cycl})$ 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 (Added: or rather, a free profinite group).
References for Morava's thoughts are
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Toward a fundamental groupoid for the stable homotopy category Link is to the arxiv, last updated 2009. There is a journal version from 2007.
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To the left of the Sphere Spectrum, from a talk given at Haynes Miller's 60th birthday conference in Bonn in 2008.
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A theory of base motives 2009. A follow-up to the previous paper according to the introduction.
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!!
Best Answer
(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:
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.