There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$: $$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \overline{\mathbb{C}\mathbb{P}^2} \right) \right) \cong \left(\mathop{\#}^{m+b^+_2(M)} \mathbb{C}\mathbb{P}^2 \right) \# \left(\mathop{\#}^{m+b^-_2(M)} \overline{\mathbb{C}\mathbb{P}^2} \right)$$ The $b^\pm_2(M)$ are the positive and negative eigenvalues of the intersection form. This has far-reaching consequences, for example any 4-manifold invariant that is multiplicative under direct sum $\#$ will only measure the intersection form on simply connected manifolds.

Is there a similar statement for non-simply-connected 4-manifolds? What is the closest we can get?