Steenrod Algebra - Cohomology Operations

Cohomology Operations

A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring, the cup product squaring operation yields a family of cohomology operations:

Cohomology operations need not be homomorphisms of graded rings, see the Cartan formula below.

These operations do not commute with suspension, that is they are unstable. (This is because if Y is a suspension of a space X, the cup product on the cohomology of Y is trivial.) Norman Steenrod constructed stable operations

for all i greater than zero. The notation Sq and their name, the Steenrod squares, comes from the fact that Sqn restricted to classes of degree n is the cup square. There are analogous operations for odd primary coefficients, usually denoted Pi and called the reduced p-th power operations. The Sqi generate a connected graded algebra over Z/2, where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case p > 2, the mod p Steenrod algebra is generated by the Pi and the Bockstein operation β associated to the short exact sequence

In the case p=2, the Bockstein element is Sq1 and the reduced p-th power Pi is Sq2i.