Some computations of stable twisted homology for mapping class groups

Abstract

In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this paper.

Communicated by Jason P. Bell

Keywords:

• Stable homology
• mapping class groups
• functor categories
• polynomial functors

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20J05
• 20J06
• 20F38
• 18D10

Acknowledgments

The author wishes to thank most sincerely Christine Vespa, Nathalie Wahl and Geoffrey Powell, for their reading, suggestions and advice. He would also like to thank Aurélien Djament, Nariya Kawazumi and Antoine Touzé for the attention they have paid to his work and helpful discussions. Additionally, he would like to thank the anonymous referee for his careful reading, comments and corrections.

Additional information

Funding

This work was partially supported by the ANR Project ChroK, ANR-16-CE40-0003 and by the JSPS Postdoctoral Fellowship Short Term PE17042.