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Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$. This is fairly easy to prove using the fact that $\mathbb{C}\mathbb{P}^n$ is monotone itself. But it is also mentioned that the positive generator of the abelian subgroup $[\mu|_{\pi_2(P,L)}]\subset \mathbb{Z}$ is $\Sigma_{\mathbb{R}\mathbb{P}^n}=n+1$. However I am not sure how one can check this last result and the author does not seem to provide any references for it.

So my question is if anyone knows any references for this or how can one go about proving this result ? Thanks in advance.

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The relative homology long exact sequence puts this group in between $H_2(\mathbb{CP}^n)=\mathbb{Z}$ and $\mathbb{Z}/2$. It maps surjectively to $\mathbb{Z}/2$. Let D be a generator for relative homology. Then $2D$ generates $H_2(\mathbb{CP}^n)$. The Maslov number of a relative class coming from $H_2(\mathbb{CP}^n)$ is twice the Chern class; for the positive generator $2D$ that gives $2(n+1)$. Therefore the Maslov number of D is $n+1$.

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  • $\begingroup$ Alright thanks, to relate the relate homology group with the group $\pi_2(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$ are you using the Hurewicz theorem ? $\endgroup$
    – user174565
    Jun 24, 2021 at 9:41
  • $\begingroup$ And also one can deal with the case where $n=1$, here we will have that $H_2(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)=\mathbb{Z}\bigoplus \mathbb{Z}$. $\endgroup$
    – user174565
    Jun 24, 2021 at 9:54
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    $\begingroup$ Yes, this is relative Hurewicz. $\endgroup$ Jun 24, 2021 at 17:42
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    $\begingroup$ When $n=1$, take the two hemispheres. These generate relative $H_2$ and you know they have the same Maslov index by symmetry. Their sum is $S^2$, whose Chern class is 2, so they each need to have Maslov 2. $\endgroup$ Jun 24, 2021 at 17:43

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