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Handlebodies separate elements of surface groups
Abstract
In this note I prove that every non-trivial element of the fundamental group of a closed orientable surface X is non-trivial in the fundamental group of some handlebody with boundary X. This result appears to be folklore. I have uploaded a proof as the result is needed in a joint project with Clement Dupont on mapping class group actions on configuration spaces.
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BibTeX
@misc{hain2026handlebodies,
title = {{Handlebodies separate elements of surface groups}},
author = {Richard Hain},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40987},
eprint = {2610.40987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40987},
}
RIS
TY - UNPB
TI - Handlebodies separate elements of surface groups
AU - Richard Hain
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40987
UR - https://arxiv.org/abs/2610.40987
AB - In this note I prove that every non-trivial element of the fundamental group of a closed orientable surface X is non-trivial in the fundamental group of some handlebody with boundary X. This result appears to be folklore. I have uploaded a proof as the result is needed in a joint project with Clement Dupont on mapping class group actions on configuration spaces.
ER -