## Algebraic & Geometric Topology

### A motivic Grothendieck–Teichmüller group

Ismaël Soudères

#### Abstract

We prove the Beilinson–Soulé vanishing conjecture for motives attached to the moduli spaces $ℳ 0 , n$ of curves of genus $0$ with $n$ marked points. As part of the proof, we also show that these motives are mixed Tate. As a consequence of Levine’s work, we thus obtain a well-defined category of mixed Tate motives over the moduli space of curves $ℳ 0 , n$. We furthermore show that the morphisms between the moduli spaces $ℳ 0 , n$ obtained by forgetting marked points and by embedding boundary components induce functors between the associated categories of mixed Tate motives. Finally, we explain how tangential base points fit into these functorialities.

The categories we construct are Tannakian, and therefore have attached Tannakian fundamental groups, connected by morphisms induced by those between the categories. This system of groups and morphisms leads to the definition of a motivic Grothendieck–Teichmüller group.

The proofs of the above results rely on the geometry of the tower of the moduli spaces $ℳ 0 , n$. This allows us to treat the general case of motives over $Spec ( ℤ )$ with coefficients in $ℤ$, working in Spitzweck’s category of motives. From there, passing to $ℚ$ coefficients, we deal with the classical Tannakian formalism and explain how working over $Spec ( ℚ )$ yields a more concrete description of the Tannakian groups.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 635-685.

Dates
Revised: 31 August 2017
Accepted: 30 October 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684012

Digital Object Identifier
doi:10.2140/agt.2018.18.635

Mathematical Reviews number (MathSciNet)
MR3773734

Zentralblatt MATH identifier
06859600

#### Citation

Soudères, Ismaël. A motivic Grothendieck–Teichmüller group. Algebr. Geom. Topol. 18 (2018), no. 2, 635--685. doi:10.2140/agt.2018.18.635. https://projecteuclid.org/euclid.agt/1521684012

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