Cohomology operation

A cohomology operation is a natural transformation between cohomology functors. Thus for every continuous map $f$, its pullback commutes with the operation: $\theta (f^{\ast }a)=f^{\ast }\theta (a)$. This makes operations additional invariants of homotopy equivalence, beyond the additive groups and their cup products.

A cohomology operation is stable if it commutes with the suspension isomorphism on reduced cohomology. Ordinary cup products become zero on a suspension, but stable operations can still connect its nonzero classes. The Steenrod squares and Steenrod reduced powers are examples.

Suppose two integral cohomology groups of a space are free of rank one. An integral isomorphism can send each chosen generator only to itself or its negative. After reduction modulo $p$, naturality of a Steenrod reduced power between those groups therefore permits its coefficient to change only by a sign, even though an abstract vector-space change of basis could rescale by any nonzero field element. At $p=5$, coefficients one and two cannot be related by signs, so they obstruct an integral homotopy equivalence. This constraint is stronger than comparing the isolated mod-five Steenrod modules with arbitrary bases.

The Steenrod algebra organizes stable mod-$p$ cohomology operations. At $p=2$ it is generated by the Steenrod squares; at an odd prime it is generated by the Steenrod reduced powers and Bockstein homomorphism. Cohomology is a module over this algebra, and every continuous map induces a homomorphism preserving its action.

An index sequence obeying the displayed inequalities represents an admissible product ${\mathrm{Sq}}^{i_1}\cdots {\mathrm{Sq}}^{i_r}$. Its degree increment is the sum of the indices. The Adem relations rewrite every square monomial in terms of admissible ones, whose Steenrod excess controls universal instability.

The excess measures the minimum degree on which an admissible Steenrod monomial can be nonzero. Strict excess below $n$ picks out polynomial generators in the mod-two cohomology of $K(\mathbb{Z}/2,n)$; equality can correspond to a square, rather than a new generator.

These mod-two identities rewrite nonadmissible products of Steenrod squares. For example, ${\mathrm{Sq}}^1{\mathrm{Sq}}^1=0$, ${\mathrm{Sq}}^1{\mathrm{Sq}}^2={\mathrm{Sq}}^3$, and ${\mathrm{Sq}}^1{\mathrm{Sq}}^3=0$. In general the sum ranges over $0\le t\le \lfloor a/2\rfloor$ and coefficients are reduced modulo two.

The Steenrod squares are natural operationsFor a degree-one class $x$, ${\mathrm{Sq}}^{1}x=x^2$, and ${\mathrm{Sq}}^1$ agrees with the mod-two Bockstein.

The Cartan product formula says that total Steenrod squares and total Steenrod reduced powers preserve the cup product:It is named after Henri Cartan and is distinct from Cartan's magic formula for the Lie derivative of a differential form, named after Élie Cartan.

For odd prime $p$, the reduced powers are natural stable cohomology operations. They satisfy $P^0=1$, the Cartan formula, and the instability conditions $P^{i}a=0$ for $2i>|a|$ and $P^{i}a=a^p$ for $|a|=2i$. In particular, a degree-two class has $P^{1}x=x^p$ and no higher nonzero reduced power. Stability lets these operations detect distinctions between suspensions whose additive cohomology and cup products agree.

Choose the degree-four generator with pullback $u\mapsto x^2$ under the complex inclusion into quaternionic projective space. On infinite complex projective space, the Cartan formula gives $P^i(x^{2k})=\left(\genfrac{}{}{0ex}{}{2k}{i}\right)x^{2k+i(p-1)}$. Injectivity of the infinite-space pullback proves the displayed formula for quaternionic projective space. Restriction to ${\mathbb{HP}}^n$ sets powers above $n$ to zero. Coefficients are reduced modulo the odd prime $p$; the exponent is an integer because $p-1$ is even.