I've been working on the exercise "Find $\pi_2 (S^2 / X)$ where $X$ is the image of $S^1 \vee S^1$ under some embedding into $S^2$".
First I tried to find some helpful fibrations/cofibrations as this is kind of what we did in the lecture last week. It wasn't very successful. Then I tried a more direct approach, i.e. drawing a picture of the situation.
Now: if I embedd $X$ into $S^2$ it will somehow distinguish $X$ into $3$ discs. Now taking the quotient, it seems pretty clear to me that $S^2 / X \cong S^2 \vee S^2 \vee S^2$. Of that I know $\pi_2$ to be $Z^3$.
Is my intuition correct? And is there an easy way to see it formally, e.g. using standard homeomorphisms for spheres such as smash or suspension?