In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions.

  1. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There should be two choices up to sign.
  2. How does one deduce the ring structure from this statement?

    The cup product is by definition given by the composition $$ H^*(\Bbb CP^n) \otimes H^*(\Bbb CP) \rightarrow H^*(\Bbb CP^n \times \Bbb CP^n) \xrightarrow{\Delta} H^*(\Bbb CP^n)$$ where $\Delta$ is the diagonal map. The first map is the cross product. By Kunneth we have an isomorphism. This shows that a pair of generators is mapped to another pair of generators.

How does the last line hold?

  • $\begingroup$ $c_1$ is distinguished because it's the first Chern class of the tautological bundle. You're free to work with $-c_1$, but that's the first Chern class of the inverse tautological bundle. $\endgroup$ – Qiaochu Yuan Apr 22 '19 at 1:00
  • $\begingroup$ If you change $c_1$ to $-c_1$ does anything at all in the proof change? $\endgroup$ – Tyrone Apr 22 '19 at 8:38

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