## 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

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,

is a degree transformation of cohomology theories.

5. (Cartan Relation)

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 .