Let $n$ be a fixed positive integer and let $K$ be a simplicial complex with $N$ vertices. Suppose the $n$-th integral homology group of $K$ is
$$ H_n(K)=\mathbb{Z}^{\oplus i}\oplus (\oplus _{p \text{ prime}}\mathbb{Z}_p^{\oplus i_p}). $$
Let $p(n,K)$ be the maximal $p$ such that $i_p$ is nonzero.
Question. Is there any upper bound of $p(n,K)$, denoted as $f(N,n)$, that does not depend on $K$? What method shall I use to find such upper bounds?
