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Quilt decompositions of surfaces and Torelli group action on the extended Hatcher complex

  • Hiromasa Fuchizawa
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Abstract

We give a presentation of a groupoid of transformations of quilt decompositions of a surface, which are pants decompositions of the surface with the addition of "seams". We consider colored quilt decompositions of a surface, where one side of a quilt decomposition is colored. We also give a presentation of a groupoid of transformations of colored quilt decompositions of a surface. These presentations of such groupoids are realized by constructing connected and simply-connected two-dimensional cell complexes whose 0-cell are these up-to-isotopy structures on the surface. In the latter part of this paper, we show that Torelli group (or more precisely, a torsion-free subgroup of a mapping class group of a surface) acts on these complexes freely.

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Cite

BibTeXRIS
BibTeX
@misc{fuchizawa2026quilt,
  title = {{Quilt decompositions of surfaces and Torelli group action on the extended Hatcher complex}},
  author = {Hiromasa Fuchizawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.02974},
  eprint = {2609.02974},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02974},
}
RIS
TY  - UNPB
TI  - Quilt decompositions of surfaces and Torelli group action on the extended Hatcher complex
AU  - Hiromasa Fuchizawa
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.02974
UR  - https://arxiv.org/abs/2609.02974
AB  - We give a presentation of a groupoid of transformations of quilt decompositions of a surface, which are pants decompositions of the surface with the addition of "seams". We consider colored quilt decompositions of a surface, where one side of a quilt decomposition is colored. We also give a presentation of a groupoid of transformations of colored quilt decompositions of a surface. These presentations of such groupoids are realized by constructing connected and simply-connected two-dimensional cell complexes whose 0-cell are these up-to-isotopy structures on the surface. In the latter part of this paper, we show that Torelli group (or more precisely, a torsion-free subgroup of a mapping class group of a surface) acts on these complexes freely.
ER  -