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The rational homology of quotients of the curve complex

  • Andrew Putman
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Abstract

Let mathopModg,n be the mapping class group of a genus-g surface with n punctures and let π’žg,n be its curve complex. For a finite-index subgroup G < mathopModg,n, Boggi proved that HΜƒk(π’žg,n/G;β„š) = 0 for g+n ≫ k. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of mathopModg,n. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

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Cite

BibTeXRIS
BibTeX
@misc{putman2026rational,
  title = {{The rational homology of quotients of the curve complex}},
  author = {Andrew Putman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10775},
  eprint = {2610.10775},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10775},
}
RIS
TY  - UNPB
TI  - The rational homology of quotients of the curve complex
AU  - Andrew Putman
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10775
UR  - https://arxiv.org/abs/2610.10775
AB  - Let mathopMod_g,n be the mapping class group of a genus-g surface with n punctures and let π’ž_g,n be its curve complex. For a finite-index subgroup G < mathopMod_g,n, Boggi proved that HΜƒ_k(π’ž_g,n/G;β„š) = 0 for g+n ≫ k. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of mathopMod_g,n. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.
ER  -