I'm trying to compute the homology groups of $\mathbb S^2$ with an annular ring whose inner circle is a great circle of the $\mathbb S^2$.
Calling this space $X$, the $H_0(X)$ is easy, because this space is path-connected then it's connected, thus $H_0(X)=\mathbb Z$
When we triangulate this space, it's easy to see that $H_2(X)=\mathbb Z$.
But I've found the $H_1(X)$ difficult to discover, I don't know the fundamental group of it, then I can't use the Hurowicz Theorem. I'm trying to find this using the triangulation of it, but there are so many calculations.
I have the following questions:
1- How we can use Mayer-Vietoris theorem in this case?
2-What is the fundamental group of this space?
3- I know the homology groups of the sphere and the annulus, this can help in this case?
I need help, please
Thanks a lot.