Homology class of real projective space inside complex projective space For any $n$ we naturally have $\mathbb{RP}^n$ inside $\mathbb{CP}^n$ and I wonder which homology class in $H_n(\mathbb{CP}^n,\mathbb{Z}_2)$ is realized by $\mathbb{RP}^n$.  Since $H_n(\mathbb{CP}^n,\mathbb{Z}_2)=0$ when $n$ is odd, this question is interesting only when $n$ is even, in which case $H_n(\mathbb{CP}^n,\mathbb{Z}_2)=\mathbb{Z}_2$. My guess is that in this case $\mathbb{RP}^n$ realizes the nontrivial class in these cases. Is that true? Can we somehow use the fact that in these cases $\mathbb{RP}^n$ is not orientable?
 A: We proceed by computing the unoriented self intersection number of $\mathbb{R}P^n $ inside $\mathbb{C}P^n$. In the $n$ even case, we will show this is $1$, and hence, that $\mathbb{R}P^n$ represents the nontrivial homology class.
The unoriented self intersection number is equal to the nth Stiefel-Whitney class of the normal bundle of the inclusion, since the normal bundle controls perturbations of the embedding and the nth Stiefel-Whitney class counts mod 2 the number of zeroes of a generic section.
Let us identify what the normal bundle is:
We put a Riemannian metric on $\mathbb{R}P^n$ so that we may talk about disk bundles. We will describe a smooth way to embed the open disk bundle (of the tangent bundle) of $\mathbb{R}P^n$, so as to conclude that the normal bundle of the embedding is actually the tangent bundle, $T(\mathbb{R}P^n )$ itself.
Around a given point $x$, we may identify the fiber of the disk bundle with the $\epsilon$-ball around $x$ in $\mathbb{R}P^n$. Locally, this is described as follows (playing fast and loose with $\epsilon$): we may pick a unit vector $(x_1,\dots,x_n)$ generating $x$ and consider the unit vectors within $\epsilon$ of $x$, each of which generate lines which together give our neighborhood of $x$.
Hence, given some $x'$ in my ball, it makes sense to talk about $x-x' = (\delta_1,\dots,\delta_n) \in \mathbb{R}^n$. Now our embedding sends $x=\langle (x_1,\dots,x_n) \rangle$ to $\langle ((x_1 +0i,\dots,x_n+0i) \rangle$. So we define $\iota: \operatorname{Disk}(T(\mathbb{R}P^n)) \rightarrow \mathbb{C}P^n$ by the fiberwise formula $x' \rightarrow \langle(x_1 + \delta_1 i, \dots, x_n + \delta_n i)\rangle$. Clearly, this extends the original embedding. It is not hard to see that if $\epsilon$ is small, this is well defined and an embedding.
So we conclude the normal bundle of this embedding is $T(\mathbb{R}P^n)$, as claimed. Now we can directly calculate the nth Stiefel Whitney class. It is equal to the mod two reduction of the Euler characteristic of $\mathbb{R}P^n$. Hence, in odd dimensions it is trivial, and in even dimensions it is nontrivial.
