Steenrod Algebra

The Steenrod algebra has to do with the Cohomology operations in singular Cohomology with Integer mod 2 Coefficients. For every $n \in \Bbb{Z}$ and $i \in \{0, 1, 2, 3, \ldots\}$ there are natural transformations of Functors

$$Sq^i: H^n(\bullet;\Bbb{Z}_2) \to H^{n+i}(\bullet;\Bbb{Z}_2)$$

satisfying:

1. $Sq^i = 0$ for $i>n$.

2. $Sq^n(x) = x \smile x$ for all $x \in H^n(X,A;\Bbb{Z}_2)$ and all pairs $(X,A)$.

3. $Sq^0 = id_{H^n(\bullet;\Bbb{Z}_2)}$.

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

$$Sq^i: H^{*}(\bullet;\Bbb{Z}_2) \to H^{*+i}(\bullet;\Bbb{Z}_2)$$

is a degree $i$ transformation of cohomology theories.

5. (Cartan Relation)

$$Sq^i(x \smile y) = \Sigma_{j+k=i}Sq^j(x)\smile Sq^k(y).$$

6. (Adem Relations) For $i<2j$,

$$Sq^i\circ Sq^j(x)=\Sigma_{k=0}^{\lfloor i \rfloor} {{j-k-1}\choose{i-2k}}Sq^{i+j-k}\circ Sq^{k}(x).$$

7. $Sq^i \circ \Sigma = \Sigma \circ Sq^i$ where $\Sigma$ 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 ${\mathcal A}$, defined to be $T(F_{\Bbb Z_2}\{Sq^i: i \in \{ 0, 1, 2, 3, \ldots\} \}) / R$, where $F_{\Bbb{Z}_2}(\bullet)$ is the free module functor that takes any set and sends it to the free $\Bbb{Z}_2$ module over that set. We think of $F_{\Bbb{Z}_2}\{Sq^i: i \in \{0, 1, 2, \ldots\} \}$ as being a graded $\Bbb{Z}_2$ module, where the $i$-th gradation is given by $\Bbb{Z}_2 \cdot Sq^i$. This makes the tensor algebra $T(F_{\Bbb{Z}_2}\{Sq^i: i \in \{0, 1, 2, 3, \ldots\} \})$ into a Graded Algebra over $\Bbb{Z}_2$. $R$ is the Ideal generated by the elements $Sq^iSq^j+\Sigma_{k=0}^{\lfloor i \rfloor} {{j-k-1}\choose{i-2k}}Sq^{i+j-k}Sq^{k}$ and $1+Sq^0$ for $0<i<2j$. This makes ${\mathcal A}$ into a graded $\Bbb{Z}_2$ algebra.

By the definition of the Steenrod algebra, for any Space $(X,A)$, $H^*(X,A;\Bbb{Z}_2)$ is a Module over the Steenrod algebra ${\mathcal A}$, with multiplication induced by $Sq^i\cdot x \equiv Sq^i(x)$. With the above definitions, cohomology with Coefficients in the Ring $\Bbb{Z}_2$, $H^*(\bullet;\Bbb Z_2)$ is a Functor from the category of pairs of Topological Spaces to graded modules over ${\mathcal A}$.

See also Adem Relations, Cartan Relation, Cohomology, Graded Algebra, Ideal, Module, Topological Space