Computing some higher homotopy groups of \(S^2\)

Necessary Theorems

UCT: The following sequence is exact \begin{align*} 0 \to \operatorname{Ext} (H_{n-1}X, {\mathbf{Z}}) \to H^nX \to \hom(H_nX, {\mathbf{Z}}) \to 0 \end{align*}

and splits unnaturally as

\begin{align*} H^nX = \operatorname{Ext} (H_{n-1}X, {\mathbf{Z}}) \oplus \hom(H_nX, {\mathbf{Z}}) \end{align*}

\(H^n(X;{\mathbf{Z}})\) and \(H_n(X; {\mathbf{Z}})\) have the same rank.
\((H^nX)_{tor} \cong (H_{n-1}X)_{tor}\)
A fibration \(F \to X \to B\) induces a LES in Homotopy
A fibration \(F \to X \to B\) induces a spectral sequence with \(E_2^{p,q} = H^p(F; H_q(B))\) and \(E_\infty \implies H_*X\)
For every \(n\), there is a map \(X \to K(\pi_n X, n)\) which induces an isomorphism on \(\pi_n\) of both spaces. This can be replaced with a fibration up to homotopy, so call the fiber \(X_{(n)}\) and this yields \begin{align*}X_{(n)} \to X \to K(\pi_n X, n)\end{align*}

Todo: typeset!