Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$.

Question

Let $X$ be the figure 8 space embedded in $S^2$. Find $\pi_2(S^2/X)$.

The problem is in page 457 of "Topology and Geometry" written by Glen E. Bredon.

I think I need to use a long exact sequence of some fibration.

But, I do not know how to approach this problem.

How I can approach the problem?

Thank you.

Answer 1

If you contract the figure 8 to a point, what do you get?

The complement of the figure 8 has 3 connected components, and all of them are open discs. When you contract the figure 8, you then get these three discs "compactified" by the same point. In other words, you get a bouquet of three spheres.

So the homotopy group is $$ \pi_2(S^2 / X) \simeq \pi_2\left(\bigvee^3 S^2\right) \simeq \bigoplus^3 \pi_2(S^2) \simeq \mathbb Z^3.$$