How can I write Klein bottle as an adjunction space? I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I don't know what $f$ should be.

How can I write the Klein bottle as an adjunction space ?

Please first give me a hint.

The Klein bottle is the quotient of a square by the equivalence relation where you identify top and bottom, and left and right, but where one of those two identifications has a twist. If you look just at the boundary of the square, the identification gives you a wedge of two circles. So when you glue in the rest of the square you get an adjunction space $(S^1\vee S^1)\cup_f D^2$. 
