---
date: 2021-04-26
---

Tags: #homotopy #homotopy 

# Computing some higher homotopy groups of $S^2$

## Necessary Theorems 

- UCT: The following sequence is exact
$$
0 \to \ext(H_{n-1}X, \ZZ) \to H^nX \to \hom(H_nX, \ZZ) \to 0
$$

  and splits unnaturally as

$$
H^nX = \ext(H_{n-1}X, \ZZ) \oplus \hom(H_nX, \ZZ)
$$

- $H^n(X;\ZZ)$ and $H_n(X; \ZZ)$ 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 $$X_{(n)} \to X \to K(\pi_n X, n)$$

## A Computation: $\pi_3 S^2$

Todo: typeset!