Small Simplicial Complexes with Prescribed Torsion in Homology

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Abstract

For \(d \ge 2\) and G a finite abelian group, define \(T_d(G)\) to be the minimum number of vertices n so that there exists a simplicial complex X on n vertices which has the torsion part of \(H_{d - 1}(X)\) isomorphic to G. Here we use the probabilistic method, in particular the Lovász Local Lemma, to establish an upper bound on \(T_d(G)\) which matches the known lower bound up to a constant factor. That is, we prove that for every \(d \ge 2\) there exist constants \(c_d\) and \(C_d\) so that for any finite abelian group
$$\begin{aligned} c_d(\log |G|)^{1/d} \le T_d(G) \le C_d(\log |G|)^{1/d}. \end{aligned}$$

Keywords

Simplicial complexes Torsion Probabilistic method Lovász Local Lemma 

Mathematics Subject Classification

55U10 60C05 

Notes

Acknowledgements

The author gratefully acknowledges support by the National Science Foundation Division of Mathematical Sciences Grant NSF-DMS #1547357. The author thanks Matt Kahle for several helpful discussions in the process of writing this paper and also thanks the two anonymous reviewers who suggested several constructive revisions on an earlier draft.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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