Finite dimensional CW complex $X$ other than wedge of two spheres so that $\tilde{H}(X) \approx \mathbb{Z} \oplus \mathbb{Z} $ I have been wanting to know distinct finite dimensional CW complexes $X$ upto homotopy equivalence with the homology $$\tilde{H}(X):=\oplus_{i \in \mathbb{N}}\tilde{H}_i(X)\approx \mathbb{Z} \oplus \mathbb{Z}$$
So far, using the Mayer-Vietoris sequence, I know that a wedge sum of two spheres $\mathbb{S}^k \vee \mathbb{S}^n$ have such a property. How can I find or disprove whether or not there exists any other CW complexes with such a property ?

Assuming you mean the reduced homology, the complex projective plane $\mathbb{C}P^2$ is another space with $\tilde{H}_*(\mathbb{C}P^2) \cong \mathbb{Z} \oplus \mathbb{Z}$.  (To see that it is not a wedge of spheres, consider the cup product in cohomology.)
