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

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Abstract

The Torelli group of a closed oriented surface Sg of genus is the subgroup Ig of the mapping class group Mod Sg consisting of all mapping classes that act trivially on the homology of Sg. One of the most intriguing open problems concerning Torelli groups is the question of whether the group I3 is finitely presented. A possible approach to this problem relies on the study of the second homology group of using the spectral sequence Erp,q for the action of I3on the complex of cycles. In this paper we obtain evidence for the conjecture that H2(I3;Z) is not finitely generated and hence I3 is not finitely presented. Namely, we prove that the term E30.2 of the spectral sequence is not finitely generated, that is, the group E10.2 remains infinitely generated after taking quotients by the images of the differentials d1 and d2 Proving that it remains infinitely generated after taking the quotient by the image of d3 would complete the proof I3 that is not finitely presented.

Original languageEnglish
Pages (from-to)1060-1127
Number of pages68
JournalIzvestiya Mathematics
Volume85
Issue number6
DOIs
Publication statusPublished - Nov 2021
Externally publishedYes

Keywords

  • action of a group on a complex
  • Birman Craggs homomorphisms
  • complex of cycles
  • homology of groups
  • mapping class group
  • spectral sequence
  • Torelli group

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