Question of maps in Mayer-Vietoris sequence We obtain MV-seq. from short exact sequence
$$
0\to C_n(A\cap B) \to C_n(A)\oplus C_n(B)\to C_n(A+B)\to 0
$$
So map i wonder that map $H_n(A\cap B)\to H_n(A)\oplus H_n(B)$ maps $[a]$ to $([a],[-a])$. But for example in this case it's not quite clear why $1 \mapsto (2,-2)$. Why $1 \not\mapsto (1,-1)$? How to make argument 'wraps twice' rigorous?
And what about other maps in this sequence?

Sam Nead has the correct suggestion here. In most instances of Mayer Vietoris you actually need to compute what the inclusion map induces or even, god forbid, what the snake homomorphism gives you. Here the boundary of a mobius strip includes into each mobius strip in such a way that it retracts onto the circle going around of the center of the mobius band. Because the mobius band is a fiber bundle, this is very trivial to show.
But the homology of the boundary of the mobius band is isomorphic to the homology of a circle, so it is generated by a cycle given by triangulating the circle. However when you use the retraction of the mobius band onto its equator, you will find that this simplex includes as twice the generator of the homology group given by the meridian. This is pretty hard to see explicitly with simplicial homology, it is easier in singular cohomology. However this is very easily seen via the Hurewicz theorem, which gives a functorial isomorphism $\pi_1(S^1) \to H_1(S^1, \mathbb{Z})$.  
