For a given coefficients sequence, $0 \to Z \to Z \to Z_2 \to 0$, where the map $Z \to Z $ is defined as times 2. My question is how to calculate the Bockstein homomorphism for real projective plane and Klein bottle? Thanks!
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This is really a question on homological algebra. I'd recommend Rotman's book "Introduction to Homological Algebra", especially Theorem 6.10 and the succeeding example in the second edition. – Tyrone Mar 18 '18 at 10:51
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@Tyrone thanks. I know. It’s just a diagram-chasing – Jiabin Du Mar 18 '18 at 10:56
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That's how to calculate it. You can also just use the fact that $\times 2:\mathbb{Z}\rightarrow\mathbb{Z}$ induces multiplication by 2 on homology groups. Then, if you know the integral and mod 2 cohomology rings of $\mathbb{R}P^2$, the fact you get a long exact sequence gives you the answer for free. You can then use the relation of $\mathbb{R}P^2$ with the Klein bottle to calculate it there. – Tyrone Mar 18 '18 at 11:09