Cohomology operation

Remarks

In the language of categories, a cohomology operation is a natural transformation between the functors $H^n(-; G)$ and $H^m(-; H)$ to sets. The functions $\theta_X$ need not be homomorphisms. Operations of type $(G, n; H, m)$ correspond bijectively to elements of $H^m(K(G, n); H)$, through $\theta \mapsto \theta(\iota_n)$, where $\iota_n \in H^n(K(G, n); G)$ is the fundamental class of the Eilenberg–MacLane space.

Example: the cup square $\theta_X(x) = x \smile x$ is an operation of type $(\Z, n; \Z, 2n)$. The check: $f^*(x \smile x) = f^* x \smile f^* x$, because $f^*$ is a ring homomorphism. It is not additive in general: $(x + y)^2 = x^2 + x y + y x + y^2$.

Non-example: on $H^0(-; \Z)$, let $\theta_X(x) = x$ if $X$ is path-connected and $\theta_X(x) = 0$ otherwise. Take $X = \set{a, b}$ discrete, $Y$ a point, $f$ the map to $a$, and $x = (1, 1) \in H^0(X; \Z) \cong \Z^2$. Then $f^* \theta_X(x) = 0$ but $\theta_Y(f^* x) = 1$, so the naturality condition fails.