2

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?

Eric Wofsey
  • 330,363
Intuition
  • 3,269
  • Which diameter? As it currently reads, it sounds like it is the union of a sphere together with a circle in the sphere, which is just the sphere itself. – J. Moeller Oct 03 '19 at 16:34
  • 3
    $X$ is homotopy equivalent to the wedge sum of $S^1$ and $S^2$. – Ayman Hourieh Oct 03 '19 at 16:50
  • @AymanHourieh can I take the open sets the northern hemisphere but slightly extending it below the equator (U) and the southern hemisphere but slightly extending it above the equator (V) ? – Intuition Oct 03 '19 at 17:42

2 Answers2

2

Use of van Kampen's Theorem

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.

Batominovski
  • 49,629
  • Could you please look at this question (if you do not mind) https://math.stackexchange.com/questions/3379198/attaching-a-disk-d2-along-the-boundary-circle-to-a-circle-s1? – Emptymind Oct 06 '19 at 11:28
1

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.

William Elliot
  • 17,598