= Cohomology ring of the complement of even-dimensional real projective space
{title2=$H^*(U;\mathbb F_2)\cong\mathbb F_2[x,y]/(x^k,y^2)$}
For $k\geq1$, put $U=\mathbb{CP}^{2k}\setminus\mathbb{RP}^{2k}$. Its <cohomology ring> is
$$
H^*(U;\mathbb F_2)\cong\mathbb F_2[x,y]/(x^k,y^2),\qquad |x|=2,\quad |y|=2k.
$$
Here $x$ 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 $0,2,\ldots,4k-2$, and zero otherwise. Choose $y$ with $\delta y=a$. The connecting-map module identity gives $\delta(x^j y)=a^{2j+1}$ for $0\leq j<k$, 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.
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