2-Complexes with Large 2-Girth
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454_2017_9926_ReferencePDF.pdf
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Author(s)
Dotterrer, Dominic
Kahle, Matthew
Guth, Lawrence
Date Issued
September 2017
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Dotterrer, Dominic, et al. “2-Complexes with Large 2-Girth.” Discrete & Computational Geometry, vol. 59, no. 2, Mar. 2018, pp. 383–412.
Version
Author's final manuscript
Abstract
The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in H[subscript 2](X,Z/2) . We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m=n[superscript 2+α] for α<1/2 , then we show that the 2-girth is at most 4n[superscript 2−2α] and we prove the existence of complexes with 2-girth at least c[subscript α,ϵ]n[superscript 2−2α−ϵ]. On the other hand, if α>1/2, the 2-girth is at most Cα . So there is a phase transition as α passes 1 / 2. Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with v vertices and f faces. Keywords: Random simplicial complexes, Homology, Counting surfaces
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00454-017-9926-3
