Determine with justification, whether $S^2 \times S^4$ is homeomorphic to $\mathbb{C}P^{3}$.
I know that they are not, but I do not know how to justify it , I got a hint that squaring a generator in $S^2 \times S^4$ is zero but in $\mathbb{C}P^{3}$ is not . could anyone help me in understanding this hint please?
If you know something about homotopy theory, you should know that $\pi _4(CP^3)=0$. To prove this, consider the fibration $S^7\to CP^3$, and look at the long exact sequence on homotopy. On the other hand $\pi _4(S^4)=\mathbb Z\neq 0$. So $S^2\times S^4$ contains a 4 sphere which is not homotopic to $0$, unlike $CP^3$