Introduction#
Our goal is to show that $$ \mathcal{A} = \text{colim } H^{n+\ast}(K(Z_2,n);Z_2)$$
Let, $\iota_n$ be the fundamental class we define a map $\mathcal{A}\to H^{\ast}(K(Z_2,n),Z_2)$ by $Sq^I \mapsto Sq^I(\iota_n)$. We will show this is an isomorphism for degree $< 2n$.
For this we will use Serre spectral sequence. There is nothing very deep about Serre spectral sequences, one can look at this for introductory parts and a few applications. The main take out from that would be the following theorem,
Serre Spectral Sequence: For a fibration $F \hookrightarrow E \to B$ there is a Spectral sequence $E^{p,q}$ with with the $E_2$ page looks like $$E_2^{p,q}= H^p(B;H^q(F))$$ and this spectral sequnce converges to $H^{p+q}(E)$.
- There is a special differential $\tau = d_n : E_n^{0,n-1}\to E^{n,0}_n$ is called the transgression.
- If $d_i(x)=0$ for $i<n$ and $d_n(x)=\tau(x)\neq 0$ then call $x$ transgressive.
- The steenrod squares commutes with the transgression.