On a spectral sequence for the action of the Torelli group of genus on the complex of cycles

The Torelli group of a closed oriented surface of genus is the subgroup of the mapping class group consisting of all mapping classes that act trivially on the homology of . One of the most intriguing open problems concerning Torelli groups is the question of whether the group is finitely presented. A possible approach to this problem relies on the study of the second homology group of using the spectral sequence for the action of on the complex of cycles. In this paper we obtain evidence for the conjecture that is not finitely generated and hence is not finitely presented. Namely, we prove that the term of the spectral sequence is not finitely generated, that is, the group remains infinitely generated after taking quotients by the images of the differentials and . Proving that it remains infinitely generated after taking the quotient by the image of would complete the proof that is not finitely presented.