Homology of the Klein Bottle I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also $H_{n}(X) = 0$ for all $n>0$ (e.g. all $1$-cycles are $1$-boundaries, etc..). How do you use this geometric interpretation to deduce that $H_{1}(X) = \mathbb{Z} \times \mathbb{Z}_2$ where $X$ is the Klein bottle? This doesn't seem to correspond to the Betti number. 

Think of the bottle as two Möbius strips glued along their edges. A closed loop that circles one of the bands (along its center, say) is not a boundary, but if you follow it twice it becomes one: you can think of it as a loop that follows precisely the edge of one of the original bands, so it's just the boundary of that band. This gives you the "torsion" piece of the homology, and it shows you that homology is a slightly more refined invariant than just counting the number of closed "empty" loops. 
EDIT: perhaps worth to point out that the group-theoretic term "torsion", which corresponds to elements of finite order, is derived precisely from this geometric picture: the torsion element result from the "twist" in the band.
