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

Hiromasa Fuchizawa

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arXiv · 2609.02974

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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Hiromasa Fuchizawa. 2026-09-02. Quilt decompositions of surfaces and Torelli group action on the extended Hatcher complex. https://arxiv.org/abs/2609.02974

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