Small simplicial complexes with prescribed torsion in homology

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 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 $c_d(\log \mathrm{\mid }G\mathrm{\mid }{)}^{1\mathrm{/}d}\le T_d(G)\le C_d(\log \mathrm{\mid }G\mathrm{\mid }{)}^{1\mathrm{/}d}\mathrm{.}$