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I'm having trouble with hatcher, page 151, namely the homology group of Klein bottle pi.math.cornell.edu/~hatcher/…. How in the world does $\Phi(1)=(2,-2)$ and what in the world does that have to do with "since the boundary circle of a Möbius band wraps twice around the core circle."

I don't understand what $\Phi$ has to do with the boundary circle wrapping twice around the core circle and what generators represent geometrically.

Arctic Char
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  • This would be helpful for geometric understanding https://math.stackexchange.com/a/315006/668308 – one potato two potato Nov 16 '21 at 18:19
  • @love_sodam Thanks. That was helpful. I'm still confused by the generator stuff –  Nov 16 '21 at 18:29
  • Mobius band is homotopy equivalent to $S^1$ (deformation retract to the core circle). $A\cap B$ in the M-V sequence is the identified boundary circle. The image of this boundary circle on deformation retracted Mobius band, which is the core circle, warps the core circle twice. – one potato two potato Nov 16 '21 at 18:36
  • @love_sodam sure, but why is that a generator and what that have to do with generators of the first homology group? –  Nov 16 '21 at 18:45
  • You can consider $\Delta$-complex. See Hatcher example 2.2 – one potato two potato Nov 16 '21 at 18:55
  • I'm trying to understand this with singular homology. Could you please write an answer please? –  Nov 16 '21 at 19:02

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