Suppose that two connected topological spaces, say $X $, $Y $, have the isomorphic fundamental groups. Is it true in general that, given two points $p\in X $, $q\in Y $, the fundamental groups of $X/\{p\} $ and $Y/\{q\} $ are isomorphic?
This question comes from this small exercise: given a function $f\in C^\infty $, defined from $S^3$ to $\mathbb {CP^1} $, show that a function $g\in C^\infty $ such that $f○g =\mathrm {Id}_{\mathbb {CP^1}}$ cannot exist. First off, the previous step of the exercise required to calculate the fundamental group of $\mathbb {CP^1} $ minus two points and the fundamental group of $S^3$ minus two points; plus, in the lectures we proved that $S^2$ and $S^3$ aren't homeomorphic by showing that $S^2$ minus two points and $S^3$ minus two points have different fundamental groups.
Now, if the answer to the question above is yes, if such a $g $ exists, $\mathbb {CP^1} $ and $S^3$ are homeomorphic, and so the fundamental groups should be isomorphic. However since the fundamental group of $\mathbb {CP^1} $ minus two points and the fundamental group of $S^3$ minus two points are respectively $\mathbb Z $ and $\{0\} $, this is absurd. If the answer is no, I would like to know why the example about $S^2$ and $S^3$ makes sense, and maybe a hint for the exercise. Thanks in advance.