What is the homology group of the sphere with an annular ring? 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 $\mathbb S^2$.
space X
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.

If I understand your space correctly, it seems you could do a deformation retraction onto $S^2$, and hence $H_1(X)=H_1(S^2)=0$.
