Consider the space $$M=\{(x_1,x_2,x_3):x_1 ^2+x_2 ^2+x_3 ^2=4\}\cup \{(x_1,x_2,x_3):x_1 ^2+x_2 ^2=1,−2≤x_3≤2\}.$$ So $M$ is basically a surface formed by a sphere of radius $2$ centered at $(0,0,0)$ along with a cylinder of height $4$ and radius $1$.
I am interested in calculating $π_1(M)$ using van Kampen's theorem. Although I can visualize what is going on, I pretty much stuck on on how to proceed.
Any help will be very useful. Thanks in advance.
Consider the Torus $T^2$ with generators $a$ and $b$ of its fundamental group (i.e. $a$ and $b$ are just the two loops in $T^2$). Let $X$ denote the space $T^2/\{a\}$. You can try to show (using pictures and visual arguments) that your space $M$ is homotopy equivalent to the space $S^2 \vee X$. Since $X$ is just the sphere $S^2$ with poles identified, we have that $X \simeq S^2 \vee S^1$, which Hatcher shows in his book in example 0.8. Then it follows that $$ \pi_1(M) \cong \pi_1(S^2 \vee X) \cong \pi_1(S^2 \vee S^2 \vee S^1) \cong \pi_1(S^1) \cong \mathbb{Z}, $$ where you can use van Kampen for the third isomorphism. I hope this helps!