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

The Shafarevich Conjecture states: $Gal(\bar Q / Q_{cyl})$ 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 (it's easy to see it is also profinite).

References for Morava's thoughts are [here] (http://folk.uio.no/rognes/yff/morava.pdf) and [here] (http://www.math.uiuc.edu/K-theory/0654/gamma2.pdf) and [here] (http://arxiv.org/PS_cache/math/pdf/0509/0509001v2.pdf) and [here] (http://www.ruhr-uni-bochum.de/topologie/conf08/jack.pdf).

This is exciting material, but I'm having trouble coming up with a way to summarize the jist 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!!