Calculating the homotopy groups of a complex

Question

I'm trying to compute the homotopy groups of the complex obtained by gluing two Klein bottles along the generator that preserves orientation. It's not dificult to compute the fundamental group, however - Is there some easy approach for the higher homotopy groups? As far as I know, it's generally hard to compute them in case they are nontrivial, but maybe there's something about this space I'm missing.

Answer 1

The universal cover of your complex is contractible and so the higher homotopy groups vanish.

To visualise it: for each of the Klein bottle take its double cover by a torus; in this way you get a double cover of your complex by two tori glued along two circles (two circles on each torus). Now take the cover of each torus by a cylinder, so that the circles lift to straight lines; we get a cover by two cylinder glued along two straight lines. But this space is homotopy equivalent to two circles glued at two points, its universal cover is a tree, and so it is contractible (the univ. cover of the original space is that tree times $\mathbb R$).