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


 - [A theory of base motives](https://arxiv.org/abs/0908.3124) 2009.


 - [The motivic Thom isomorphism](https://arxiv.org/abs/math/0306151) 2003. 


 - [Toward a fundamental groupoid for the stable homotopy category](https://arxiv.org/abs/math/0509001) Link is to the arxiv, last updated 2009. There is a [journal version](https://msp.org/gtm/2007/10/gtm-2007-10-017p.pdf) from 2007.

 - [To the left of the Sphere Spectrum](http://www.ruhr-uni-bochum.de/topologie/conf08/jack.pdf), 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!!