Calculating $\pi_{1}(X).$

Question

If $X$ is the union of $S^2$ and the diameter between the North and South poles. Calculating $\pi_{1}(X)$ using Van Kampen theorem.

Could anyone give me a hint about the open sets I should use?

Answer 1

In the figure, the space $X$ is given by the green ball with the blue line on the left. Now I let $U$ be the red shell and the blue line on the right, and $V$ the orange shell. Observe that $U$ is homotopic to $S^1$ and $V$ is contractible, while $U\cap V$ is homotopic to $S^1$. By van Kampen's theorem, $$\pi_1(X)\cong \pi_1(U)\underset{\pi_1(U\cap V)}{*}\pi_1(V).$$ But since any loop in $U\cap V$ is null-homotopic in $U$, the map $\pi_1(U\cap V)\to \pi_1(U)$ is trivial (the map $\pi_1(U\cap V)\to \pi_1(V)$ is also trivial because $\pi_1(V)$ is trivial), so $$\pi_1(X)\cong \big(\pi_1(U)*\pi_1(V)\big)/\{1\}\cong\big(\Bbb Z*\{1\}\big)/\{1\}\cong\Bbb Z/\{1\}\cong\Bbb Z.$$ Note that $\pi_1(X)$ is generated by a loop $g$ that passes through the diameter $NS$ exactly once.

Answer 2

A base for the space is
the open sets of the sphere that do not contain the poles
the open sets of the diameter that do not contain the poles
the open sets of the sphere that contain just one pole and an open ended line segment from the pole along the diameter.