Cohomology ring of complex projective space ($H^{\ast }({\mathbb{CP}}^n;\mathbb{Z})$)

If $x=c_1(\mathcal{O}(1))$ has degree two, thenReducing coefficients modulo two gives

Give Complex projective space its complex orientation. In the homology cross product basis of $H_{2n}({\mathbb{CP}}^n\times {\mathbb{CP}}^n;\mathbb{Z})$, the diagonal has the displayed class. If $a,b$ are the two normalized degree-two hyperplane classes, its Poincare dual is ${\sum }_{j=0}^n{a}^j{b}^{n-j}$. Indeed every degree-$2n$ monomial $a^r{b}^{n-r}$ restricts on the diagonal to the top power of the hyperplane class, whose evaluation on the fundamental class is one. These evaluations determine all coefficients.

Give $(S^2{)}^m$ its product orientation and ${\mathbb{CP}}^m$ its complex orientation. If $u_i$ are the degree-two sphere classes and $h$ the normalized projective class, write $f^{\ast }h={\sum }_i{a}_i{u}_i$. In the source cohomology ring, $u_i^2=0$ and the generators commute. Hence $f^{\ast }{h}^m=m!({\prod }_i{a}_i)u_1\cdots u_m$. Evaluation on the fundamental class gives the displayed mapping degree. Every positive degree is therefore at least $m!$, and the polynomial multiplication map to complex projective space attains that bound.

Let $i:{\mathbb{RP}}^d\hookrightarrow {\mathbb{CP}}^d$ be the standard inclusion. If $h=c_1(\mathcal{O}(1))\mathrm{mod}2$ and $a=w_1({\gamma }_d)$, thenThe restricted complex tautological bundle is the complexification of a real vector bundle ${\gamma }_d{\otimes }_{\mathbb{R}}\mathbb{C}$. Its underlying real bundle is ${\gamma }_d\oplus {\gamma }_d$. The Whitney product formula for Stiefel–Whitney classes gives total class $(1+a{)}^2=1+a^2$, while the Stiefel–Whitney class of the underlying real bundle of a complex line identifies its second class with the reduced First Chern class. Dualizing changes the integral sign but has no effect modulo two. Thus the full induced map on cohomology sends $h^j$ to $a^{2j}$.

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.

For $n\ge 1$, every graded-ring automorphism ofacts on $H^2$ by a signed permutation matrix. All eight such matrices are realized by independently applying complex conjugation to the factors and by interchanging the factors.

For $0<k<n$, let $X={\mathbb{CP}}^n/{\mathbb{CP}}^k$ and let $x_i\in H^{2i}(X;\mathbb{Z})$ pull back to $x^i\in H^{2i}({\mathbb{CP}}^n;\mathbb{Z})$. Additively,and its positive-degree multiplication is