I learned from Hatcher's book Algebraic Topology Section 4.C that we can know the mininal cell structure of a simply connected CW complex from its homology groups. The theorem is as follows:
Proposition 4C.1. Given a simply-connected CW complex $X$ and a decomposition of each of its homology groups $H_n(X)$ as a direct sum of cyclic groups with specified generators, then there is a CW complex Z and a cellular homotopy equivalence $f:Z\to X$ such that each of $Z$ is either:
(a) a 'generator' $n$-cell $e_{\alpha}^n$, which is a cycle in cellular homology mapped by $f$ to a cellular cycle representing the specified generator $\alpha$ of one of the cyclic summands of $H_n(X)$; or
(b) a 'relator' $(n+1)$-cell $e_{\alpha}^{n+1}$, with cellular boundary equal to a multiple of the generator $n$-cell $e_{\alpha}^n$, in the case that $\alpha$ has finite order.
I wonder if there is a similar result for relative CW complexes?