Small simplicial complexes with prescribed torsion in homology
Abstract
For d ≥ 2 and G a finite abelian group, define Td(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 Hd - 1(X) isomorphic to G. Here we establish an upper bound on Td(G) which matches the known lower bound up to a constant factor. That is, we prove that for every d ≥ 2 there exist constants cd and Cd so that for any finite abelian group cd( |G|)1/d ≤ Td(G) ≤ Cd( |G|)1/d.
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