Intuition behind homology of a Klein bottle

Question

I understand the calculations behind the homology of a Klein bottle, but am struggling with the intuition of what it means. I know that $H_1 = \mathbb{Z} \oplus \mathbb{Z}_2$ but am confused on what that would actually correspond to, specifically the $\mathbb{Z}_2$. Could someone please explain?

Answer 1

Think of the Klein bottle as a cylinder S¹×[0,1] with its two boundary circles S¹×{0} and S¹×{1} identified to each other by the homeomorphism h : S¹×{0} → S¹×{1} defined via h((𝜃,0)) = (-𝜃,1).

Then in homology (with integer coefficients) these two circles are equal to each other (as boundary circles of the same cylinder) but they are also negatives of each other (because of the identification). Therefore they generate the group ℤ₂.