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Mathematics > Geometric Topology

arXiv:1512.08263 (math)
[Submitted on 22 Dec 2015 (v1), last revised 12 Nov 2016 (this version, v2)]

Title:Invariants and TQFT's for cut cellular surfaces from finite groups

Authors:Diogo Bragança, Roger Picken
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Abstract:We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a "flatness" condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions.
Comments: 28 pages, 27 figures. Revised version, including a topological proof of the property: the number of conjugacy classes of a finite group G equals the commuting fraction of G times the order of G. To appear in Boletim da Sociedade Portuguesa de Matemática
Subjects: Geometric Topology (math.GT)
MSC classes: 57M20, 57M50
Cite as: arXiv:1512.08263 [math.GT]
  (or arXiv:1512.08263v2 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.1512.08263
Journal reference: Boletim da Sociedade Portuguesa de Matemática, 74 (2016), p. 17

Submission history

From: Roger Francis Picken [view email]
[v1] Tue, 22 Dec 2015 21:08:33 UTC (61 KB)
[v2] Sat, 12 Nov 2016 12:05:46 UTC (61 KB)
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