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 (This is a broken link to what appears to be an old website of John Rognes, who is a user here and in fact left the existing partial answer when this question was originally asked in 2010.)
Toward a fundamental groupoid for the stable homotopy category Link is to the arxiv, last updated 2009. There is a journal version from 2007.
To the left of the Sphere Spectrum, from a 2008 conference proceedings judging by the URL.
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!!
