Section 23.5 Serre’s Finiteness Theorem

The case \(n = 1\) is well-known (\(\pi_i(S^1) = 0\) for \(i \geq 2\)). Assume \(n \geq 3\) odd. Let \(f \colon S^n \to K(\mathbb{Z}, n)\) generate \(\pi_n K(\mathbb{Z}, n) \cong \mathbb{Z}\text{,}\) and replace \(f\) by an equivalent fibration \(p \colon E_f \to K(\mathbb{Z}, n)\) with \(S^n \xrightarrow{\simeq} E_f\text{.}\) Let \(F\) be the fibre. The long exact sequence in homotopy gives:

So \(F \simeq S^n_{> n}\) (the \(n\)-connected cover of \(S^n\)), and it suffices to show \(\pi_q(F)\) is finite for all \(q > n\text{.}\) By repeated application of Hurewicz + Whitehead-tower arguments (Theorem 23.5 and its proof), it is enough to show \(H_q(F; \mathbb{Z})\) is finite for all \(q > n\text{.}\)

Claim: \(H_q(F; \mathbb{Z})\) is finite for all \(q > n\text{.}\)

Suppose for contradiction that there is some \(q > n\) with \(H_q(F)\) infinite, and let \(v > n\) be the minimal such. Then \(H_v(F)\) has positive rank, while \(H_q(F)\) is finite for all \(q\) with \(n < q < v\text{.}\)

For odd \(n\text{,}\) the cohomology \(H^*(K(\mathbb{Z}, n); \mathbb{Q})\) is an exterior algebra on a single generator in degree \(n\text{;}\) in particular \(H^q(K(\mathbb{Z}, n); \mathbb{Z})\) is finite for \(q > n\) except in degree \(0\text{.}\) By UCT, \(H_q(K(\mathbb{Z}, n); \mathbb{Z})\) is finite for \(q > 0\) with \(q \neq n\text{,}\) and \(H_n(K(\mathbb{Z}, n); \mathbb{Z}) = \mathbb{Z}\text{.}\)

Look at the bidegree \((0, v)\text{:}\) it contains \(E^2_{0,v} = H_v(F)\text{,}\) which is infinite. Every differential out of \(E^r_{0,v}\) vanishes, since its target \(E^r_{-r,\, v+r-1}\) lies outside the first quadrant. Every differential into \(E^r_{0,v}\) originates at \(E^r_{r,\, v-r+1}\text{,}\) and this source is always finite: if \(v - r + 1 > 0\) it is a subquotient of a finite \(E^2\)-entry by the previous paragraph, while if \(v - r + 1 = 0\) (that is, \(r = v + 1\)) it is a subquotient of \(E^2_{v+1,\,0} = H_{v+1}(K(\mathbb{Z},n))\text{,}\) which is finite because \(v + 1 > n\text{.}\) Hence the image of every incoming differential is a finite subgroup, and consequently each \(E^{r+1}_{0,v} = E^r_{0,v} / \mathrm{im}(d^r)\) has the same rank as \(E^r_{0,v}\) — infinite. By stabilisation, \(E^\infty_{0,v}\) is infinite.

But by convergence, \(E^\infty_{0,v} = F_0 H_v(S^n) \subseteq H_v(S^n) = 0\) (since \(v > n\)). Contradiction.

Hence \(H_q(F)\) is finite for all \(q > n\text{,}\) and therefore \(\pi_q(F) \cong \pi_q(S^n)\) is finite for all \(q > n\text{.}\)

Combine Theorem 23.6 with Freudenthal’s theorem: \(\pi_q^s = \pi_{q+n}(S^n)\) for \(n\) sufficiently large, in particular for some odd \(n > q + 1\text{.}\)