Consider the task:
Let $X = T \# T$, the sum of $2$ tori identified by removing a small disk from each of them and gluing the missing disks along the boundary. Find the fundamental group of $X$, show the universal cover is contractible, and deduce that any image of $S^n$ into $X$ is contractible.
As for the first part i assume one should apply Van Kampen to the spaces of the Tori with a disk removed (so that the intersection is homotopically equivalent to the $S^1$ around the removed part). Now I think the torus with a disk removed is homotopically equivalent to a torus with a point removed, which is the wedge of two circles. But one of this circles is generated by the intersection in Van Kampen, so that one we consider to be a trivial loop. In the end Van Kampen gives us that $X$ has the fundamental group of $Z * Z$ which coincedentally is again the fundamental group of a wedge of two circles.
Now my understanding of Van Kampen is dubious at best, is this line of thinking correct? And how do i go about finding the universal cover? There's a hint to the second part of the task to look at covers of $X$ homotopically equivalent to a wedge product of circles, but I'm not sure how to even define such covers. How would one proceed with the rest of the task?
As pointed out in the comments by multiple people:
The original line of thinking about the fundamental group makes no sense and comes from a misinterpreted van Kampen. The proper answer given by the theorem is the group $<a,b,x,y| [a,b] = [x,y]>$.
What remains is the part about the universal cover