José São João: Koszulness of Torelli Lie algebras

Abstract.

In this thesis we will follow the paper “On the Torelli Lie algebra” by Kupers and Randal-Willaims [40] to prove that the Lie algebra Gr_{lcs}^* t_{g,1} is Koszul. The Lie algebra Gr_{lcs}^* t_{g,1} is associated to the Torelli group of a surface and proving the Koszulness answers a conjecture of Richard Hain [21]. We will introduce the auxiliary objects Zn and En. Using category theory and the the- ory of Sp2g(Z)-representations we will derive the connection between E1 and the quadratic dual of Gr_{lcs}^* t_{g,1} which will allows us to study the Koszulness of Gr_{lcs}^* t_{g,1} using E1. Finally using results on high-dimensional manifolds and results about graphs complexes we prove that Zn and En are Koszul implying that Gr_{lcs}^* t_{g,1} is Koszul in a stable range.