Klein bottle
The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a CW complex structure with one 0-cell "P", two 1-cells "C"1, "C"2 and one 2-cell "D". Its Euler characteristic is therefore . The boundary homomorphism is given by and , yielding the homology groups of the Klein bottle "K" to be , and for .