I need to calculate the fundamental group of the 2-sphere with 2 disks removed and then attached 2 Mobius Strips along the boundary of the disks using Seifer Van Kampen
I started by first attaching only one Mobius Strip. Take $U$ an open neighborhood deformation retracting to the Mobius Strip, and its fundamental group is $\mathbb Z$. Then take an open neighborhood $V$ deformation retracting to the 2-sphere with a disk removed, and its fundamental group is trivial. The intersection of $U$ and $V$ is homotopic to a circle thus having a fundamental group being $\mathbb Z$. Then the resulting fundamental group of $U \cup V$ should be $\mathbb Z$ quotient by itself. So it is trivial.
Now, take $U$ to be an open neighborhood deformation retracting to the Mobius Strip with one mobius strip attached, and its fundamental group is $\mathbb Z$ because it deformation retractes to the boundary of the mobius strip. Then take an open neighborhood $V$ deformation retracting to the other Mobius Strip, and its fundamental group is $\mathbb Z$. The intersection of $U$ and $V$ is homotopic to a circle thus having a fundamental group being $\mathbb Z$. The resulting fundamental group is $\mathbb Z \ast \mathbb Z$ quotient out one copy of the $\mathbb Z$. So the final result is $\mathbb Z$.
But this does not make sense because there are at least two nontrivial loops, one surrounding each mobius band. On second thought it does make sense since we can take the loop around one of the mobius band and "pull it down along the sphere" then it will be a loop surrounding the other mobius band.