$\DeclareMathOperator\Q{\mathbf{Q}}$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

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 talk given at Haynes Miller's 60th birthday conference in Bonn in 2008.

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 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!!

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