Let $X$ be the space obtained by attaching the boundary of a disc to the meridian of a torus. I am trying to compute $\pi_2(X)$.
So far, I showed that $X$ is homotopic to $S^1 \vee S^2$, so the problem reduces to computing $\pi_2(S^1 \vee S^2)$. Moreover, let $Y$ be the space consisting of a real line with a sphere attached at each integer points. Then $Y$ is a covering space of $S^1 \vee S^2$, and so $\pi_2(S^1 \vee S^2) \cong \pi_2(Y)$. But, I am sure how to compute $\pi_2(Y)$.
I would appreciate any help, hint or reference.
Hint: $Y$ is the universal covering space of $S^1 \vee S^2$, i.e. the simply connected covering space. So the Hurewicz theorem applies, with the conclusion that $\pi_2(Y) \approx H_2(Y;\mathbb Z)$.