Hi I'm a little bit stuck on this problem
Let $$S_1 =\{ (x,y,z) \in \mathbb{R}^3: x^2+(y-1)^2+z^2=1\}\\S_2 =\{ (x,y,z) \in \mathbb{R}^3: x^2+(y+1)^2+z^2=1\}$$ consider also $$C=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2 = 4, z\in [-2,2]\}$$ and union of upper and lower cylinder's basis $$B =\{(x,y,z)\in \mathbb{R}^3: x^2+y^2 \leq 4, z =-2,z= 2\}$$ Calculate the fundamental group $\pi_1(S_1\cup S_2\cup B\cup C, p)$ where $P=(0,2,0)$
I represent everything on the cartesian plane and it's a cylinder of radius $2$ and two radius $1$ spheres touching in the origin and tangent to $C$. I've tried SVK theorem but I could not say anything useful, even though I have a feeling that this group is isomorphic to $\mathbb{Z}$. Here's why I think that (sorry for the poor drawing)
I think that every loop that goes only on the cylinder like $\alpha$ are homotopic to the constant loop, the same holds for all the loops that goes on the spheres and come back to $P$ whitout going on the cylinder. So I think the only loop that is important is $\delta$ the yellow one going on the spheres and cylinder. I honestly don't know if what I'm saying make sense and I still don't know how to calculate this fundamental group. Thanks for the help.
Your guess that the fundamental group is $\mathbb{Z}$ is correct. Here is a way to see that your space is homotopy equivalent to a wedge product of a circle and three spheres, which shows the fundamental group is $\pi_1(S^1)*\pi_1(S^2)*\pi_1(S^2)*\pi_1(S^2)=\mathbb{Z}*1*1*1=\mathbb{Z}$:
I hope it is clear that the last space is indeed equivalent to a wedge product of a circle and three spheres.