Let $X \subset \mathbb{C}^2$ be given by the equation
$$|z|^2 + |w|^2=1$$ and let $A \subset X$ be given by $|z|=1,\ w=0$.
This question requires me to find the relative homology groups $H_n(X,A)$, but before I do that I would like to ask if my guess is correct.
Is $X$ homeomorphic/homotopy equivalent to $S^4$, and $A$ homeomorphic/homotopy equivalent to $S^2$?