As an application of the theorem
(Serre) The cohomology ring $\mathcal{H}^*(K(\mathbb{Z}_2,n);\mathbb{Z}_2)$ is isomorphic to $\mathbb{Z}_2[Sq^I(\iota_n)]$ where $\iota_n$ is a fondamental class of $\mathcal{H}^n(K(\mathbb{Z}_2,n);\mathbb{Z}_2)$ and $I$ is ammissible with excess less than $n$.
I find in Mosher's book: Cohomology operations and applications in homotopy theory
The computing of $\pi_{3+k}(S^3)$ with $k\le 3$. In particular, using the map $$Sq^2(\iota_3):K(\mathbb{Z},3)\longrightarrow K(\mathbb{Z}_2,5),$$ he obtains a fibration $F_1\longrightarrow X_1\longrightarrow B$ where
- $X_1$ is the pull-back of $Sq^3(\iota_3)$ and the path-fibration $PK(\mathbb{Z}_2,5)\longrightarrow K(\mathbb{Z}_2,5)$
- $F_1=K(\mathbb{Z}_2,4)$
- $B=K(\mathbb{Z},3)$
After some calculation in the Serre spectral sequence associated to the fibration $F_1\longrightarrow X_1\longrightarrow B$, he find $\pi_{4}(S^3)$ and $\pi_{5}(S^3)$.
My question is, after computing the $H^*(X_1)$ how I can find this homotopy groups for $S^3$? I don't see this link.
Thanks all for answering
As they explain on p. 113, there is a map $S^3 \to K(\mathbb{Z}, 3)$, and the composite $$ S^3 \to K(\mathbb{Z}, 3) \xrightarrow{Sq^2} K(\mathbb{Z}/2, 5) $$ is zero, so there is a lift to a map $S^3 \to X_1$. This map induces an isomorphism on cohomology in dimensions up to 4, mono in dimension 5, so it induces an isomorphism on $\pi_4$.
Now the point is to continue this pattern: find a space $X_2$, a map $X_2 \to X_1$, and a lift $S^3 \to X_2$ which induces an isomorphism on cohomology in a larger range of dimensions. So you need to compute the cohomology of $X_1$ so as to know how to get a fibration $X_2 \to X_1 \to K(?, ?)$. Then you will know the homotopy groups of $X_2$ and they will agree with those of $S^3$ through dimension 5.