I have seen the following result in a few algebraic topology texts (such as Spanier), but only as an exercise:
For any sequence $m_1, \ldots, m_n$ of nonnegative integers, there is a connected simplicial complex $K$ with $H_p(K)$ free abelian of ran $b_p$ for every $1 \le p \le n$ (for the sake of clarity, the $H_p$ notation denotes the $p$-th homology group).
It looks really cool, and I have tried to work out a proof of it, but I am having trouble gaining momentum. I was wondering if anyone visiting today knows how to deduce this result, and if so, would be willing to prove it or give hints towards proving it. Any constructive responses would be greatly appreciated!