 Solution

This sequence does occur, for the oriented connected sumThe Künneth theorem gives one copy of $\mathbb{Z}$ in dimensions $0,1,3,4$ for each factor product and zero in dimension two. Removing a disc from each summand and gluing their boundary three-spheres, the Mayer–Vietoris sequence makes the homology in dimensions one through three the direct sum of that of the two summands. The two orientations join to one top fundamental class, so $H_0=H_4=\mathbb{Z}$, $H_1=H_3={\mathbb{Z}}^2$, and $H_2=0$.