I'm trying to understand why my calculation of this fundamental group doesn't work:
In my diagram, the first figure is a cube, with the sides identified in the obvious way. Via homotopy equivalence, we separate the faces, with lines stretching between identified vertices (we can do this as the portion of the edges between vertices have been deleted) and collapse the interior of each face to a point.
Then, via homotopy equivalence, we slide all of the loops together, which yields a wedge of $19$ loops, so I conclude that the fundamental group is the free group with $19$ generators. This is apparently incorrect.
Can anyone point out exactly where my argument fails? Thanks!