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There are similar questions to mine, but none that touch the issue I am having: On Wikipedia (Link), the Mayer-Vietoris sequence is applied to compute the singular homology of the Klein bottle. It says "the central map $\alpha$ sends $1$ to $(2,-2)$ since the boundary circle of a Möbius band wraps twice around the core circle". I can imagine why it wraps around the core circle twice, but how is this connected to the map $\alpha$? I guess the deeper question is, what the generators of homology groups explicitly look like. I haven't found any material on this.

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For the Möbius band the generator of $H_1$ is the circle which is in the middle of the band. This is for example because it deformation retracts on this circle, or from cellular homology if you represent the band as a quotient space of a rectangle.

The $\mathbb Z$-part of the sequence is generated by the path from $A$ to $B=C$ and then from $C$ to $D=A$. For the red Möbius strip this is its whole boundary, which goes around strip's middle circle twice. The same thing for the blue Möbius strip.

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