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2-Complexes with Large 2-Girth

Abstract

The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in \(H_2(X, {\mathbb {Z}}/2)\). We consider the maximum possible girth of a complex with n vertices and m 2-faces. If \(m = n^{2 + \alpha }\) for \(\alpha < 1/2\), then we show that the 2-girth is at most \(4 n^{2 - 2 \alpha }\) and we prove the existence of complexes with 2-girth at least \(c_{\alpha , \epsilon } n^{2 - 2 \alpha - \epsilon }\). On the other hand, if \(\alpha > 1/2\), the 2-girth is at most \(C_{\alpha }\). So there is a phase transition as \(\alpha \) 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.

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Acknowledgements

L.G. was supported in part by a Simons Investigator Grant. M.K. gratefully acknowledges support from DARPA #N66001-12-1-4226, NSF #CCF-1017182 and #DMS-1352386, the Institute for Mathematics and its Applications, and the Alfred P. Sloan Foundation. All three authors thank the Institute for Advanced Study for hosting them in Spring 2011, when some of this work was completed.

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Correspondence to Matthew Kahle.

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Editor in Charge: Kenneth Clarkson

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Dotterrer, D., Guth, L. & Kahle, M. 2-Complexes with Large 2-Girth. Discrete Comput Geom 59, 383–412 (2018). https://doi.org/10.1007/s00454-017-9926-3

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Keywords

  • Random simplicial complexes
  • Homology
  • Counting surfaces

Mathematics Subject Classification

  • 05C80
  • 55U10