Steenrod Algebra

The Steenrod algebra has to do with the cohomology operations in singular cohomology with integer mod 2 coefficients. For every  and  there are natural transformations of functors

(1)

satisfying:

1.  for .

2.  for all  and all pairs .

3. .

4. The  maps commute with the coboundary maps in the long exact sequence of a pair. In other words,

(2)

is a degree  transformation of cohomology theories.

5. (Cartan relation)

(3)

6. (Adem relations) For ,

(4)

7.  where  is the cohomology suspension isomorphism.

The existence of these cohomology operations endows the cohomology ring with the structure of a module over the Steenrod algebra , defined to be , where  is the free module functor that takes any set and sends it to the free  module over that set. We think of  as being a graded  module, where the th gradation is given by . This makes the tensor algebra  into a graded algebra over .  is the ideal generated by the elements  and  for . This makes  into a graded  algebra.

By the definition of the Steenrod algebra, for any space ,  is a module over the Steenrod algebra , with multiplication induced by . With the above definitions, cohomology with coefficients in the ring ,  is a functor from the category of pairs of topological spaces to graded modules over .