How can I calculate this fundamental group?

Question

I’m having some troubles with fundamental groups. How can I calculate the fundamental group of $X = S^2 \cup C$ where $C=\{x^2+y^2=1/4, z\in \mathbb{R}\}$ is the cylinder?

Thank you so much :)

Answer 1

Contracting the cylinder vertically to a circle, $X$ is homotopy equivalent to the union of a torus and a sphere glued along a circle (the equator of the sphere and a primitive loop on the torus). Now we can use van Kampen: $$\pi_1(X) \cong (\pi_1(S^1 \times S^1) * \pi_1(S^2))/\pi_1(S^1) \cong \mathbb{Z}^2/\langle1, 0\rangle \cong \mathbb{Z}.$$

Alternatively, contracting one of the halves of the sphere gets us to a sphere wedged with a pinched torus. The pinched torus is equivalent to $S^1 \vee S^2$, so $X \simeq S^1 \vee S^2 \vee S^2$ and now a simpler application of van Kampen shows $\pi_1(X) \cong \mathbb{Z}$.

The generator is given by the other factor of the torus, which before contracting is the "D" shaped loop described in the comments.