I want to find a manifold of dimension 4 which fundamental group is $F_{\alpha}$ (the free group on $\alpha$ generators). Since the fundamental group of the connected sum is the free group of the fundamental groups I think it just be enough to find a 4th dimensional manifold which fundamental group is $\mathbb{Z}$.
I don't know if $D^{3}\times S^{1}$ is a 4th dimensional manifold but it would work since $D^{3}$ (the ball ofdimension 3) is contractible.