We know that topologically $ \mathbb{R}^{2n} \simeq \mathbb{C}^n $ by, for example, $ (x_1, y_1, \dots, x_n, y_n) \mapsto (x_1 + i y_1, \dots, x_n + i y_n) $. The unit balls in two spaces, both of which are even called the same, $S^{2n-1}$, are also homeomorphic. Then, we get $ \mathbb{RP}^3 $ and $ \mathbb{CP}^2 $ by identifying antipodal points in here.
What is wrong with this argument? At least I know their fundamental groups are different, so they should not be homeomorphic. Moreover, generalizing the above argument, $ \mathbb{RP}^{2n-1} $ should be homeomorphic to $ \mathbb{CP}^n $ for every $n$, which is not the case as well. My chain of thought is just a generalization of what we do to show that $ \mathbb{RP}^1 $ is same as $ \mathbb{CP}^1 $, but I can't find what I am missing.