 Cohomology ring of the complement of even-dimensional real projective space

For $k\ge 1$, put $U={\mathbb{CP}}^{2k}\setminus {\mathbb{RP}}^{2k}$. Its cohomology ring isHere $x$ is the restricted hyperplane class. The cohomological Gysin map of an embedding sends $a^{2j}$ to $h^{k+j}$ and every odd-degree class to zero, by Poincare duality and mod-two restriction from complex to real projective space. Its exact sequence gives one-dimensional groups in even degrees $0,2,\dots ,4k-2$, and zero otherwise. Choose $y$ with $\delta y=a$. The connecting-map module identity gives $\delta (x^{j}y)=a^{2j+1}$ for $0\le j<k$, so these are the upper-half generators. The lower-half generators are $x^j$. The relations $x^k=0$ and $y^2=0$ follow respectively from $i_{!}1=h^k$ and $H^{4k}(U;{\mathbb{F}}_2)=0$. The result is a ring isomorphism with Complex projective space ${\mathbb{CP}}^{k-1}$ times $S^{2k}$; it does not by itself assert a homotopy equivalence of the spaces.