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How can I compute Cohomology of complex projective plane $CP^2$? Any magic like the one here?

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up vote 9 down vote accepted

There are a lot of different approaches depending upon what you know, what precise incarnation of cohomology you're using, and so forth.

Off the top of my head, the following seems easiest: for any $n \in \mathbb{Z}^+$, complex projective $n$-space $\mathbb{C} \mathbb{P}^n$ has the natural structure of a CW-complex with exactly one cell in each degree $2k$ for $0 \leq k \leq n$ and no cells in any other degrees. (You can see this by writing projective $n$-space as affine $n$-space union projective $n-1$ space.) This makes the co/homology work out about as simply as it gets: all the attaching maps are zero, so we have $H_i(\mathbb{C} \mathbb{P}^n, \mathbb{Z})$ is $\mathbb{Z}$ if $0 \leq i \leq 2n$ and $i$ is even and $0$ otherwise. The vanishing of all the odd homology groups also makes the Universal Coefficient Theorem easy to apply, and one gets exactly the same answer for the cohomology groups.

If you wanted to use some other method -- or, if you need to know the ring structure of the cohomology (short answer: it is a truncated polynomial ring) -- then please let me know.

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P.S.: The space $\mathbb{C} \mathbb{P}^n$ is simply connected (unlike real projective space), so I don't see how to apply the sort of "magic" used in the answer you linked to. – Pete L. Clark Mar 29 '11 at 0:12
@Pete L. Clark Unfortunately I am not familiar with $CW$-complex. – timhortons Mar 29 '11 at 0:51
@MON: Okay, then please provide some further information about your background. – Pete L. Clark Mar 29 '11 at 3:19
@Pete L. Clark I just know a bit of cohomology. – timhortons Mar 29 '11 at 3:45
$H_i(\mathbb CP^n,\mathbb Z) = \mathbb Z $ if **$0\leq i \leq 2n$** and $i$ is even. – Alexander Thumm Mar 29 '11 at 7:05

Let $f: \mathbb CP^n \to \mathbb R$ be given by: $$[z_0, \dots, z_n] \mapsto \frac{|z_1|^2 + 2|z_2|^2 + \dots + n|z_n|^2}{|z_0|^2 + |z_1|^2 + \dots + |z_n|^2}$$ One can show that $f$ is a morse function with morse polynomal $P_f(t) = \sum_{k=0}^n t^{2k}$, which satisfies the gap condition and therefore $f$ is a perfect morse function. We conclude $H^k(\mathbb CP^n,\mathbb R) = \mathbb R$ for $k = 0,2,\dots,2n$ and $H^k(\mathbb CP^n,\mathbb R) = 0$ otherwise.

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This approach is by the way in no way fundamentally different from that taken by Pete L. Clark in his answer, as morse theory gives us a CW complex with one cell in each even dimension $k\leq 2n$. – Alexander Thumm Mar 29 '11 at 12:19

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