I have a CW complex constructed as follows:
(The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein bottles and a Projective space.
I'm trying to compute its homotopy groups.
The fundamental group is clear, but for higher homotopy groups I'm a bit stuck: If I'm trying to use a universal cover, I get a space equivalent to an uncountable wedge of 2-spheres, but as far as I know there's no method for calculating the homotopy groups of that. The other tools I know deal with homotopy groups up to some degree, but that doesn't help me as I want to describe all of them.
Could anyone suggest a different approach, or let me know if I'm missing something in my original approach? Any help is much appreciated.