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 theoryandgeometric topologyby 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!!