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For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be theminimum 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 establish an upper bound on $T_d(G)$ which matches the known lowerbound up to a constant factor. That is, we prove that for every $d \geq 2$there exist constants $c_d$ and $C_d$ so that for any finite abelian group$c_d(\log |G|)^{1/d} \leq T_d(G) \leq C_d(\log |G|)^{1/d}.$

Small simplicial complexes with prescribed torsion in homology