nLab Serre finiteness theorem

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology

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There are several theorems by Serre which deserve to be called “finiteness theorems”.

In homotopy theory

On the homotopy groups of spheres:

Theorem

(Serre finiteness theorem)

The homotopy groups of spheres π n+k(S k)\pi_{n+k}(S^k), for k1k \geq 1, are finite groups (in fact finite abelian groups), hence pure torsion, except

  1. for n=0n = 0 in which case π k(S k)=\pi_k(S^k) = \mathbb{Z} is the integers;

  2. k=2mk = 2m and n=2m1n = 2m -1 in which case

    π 4m1(S 2m)F m \pi_{4m - 1}(S^{2m}) \simeq \mathbb{Z} \oplus F_m

    for F mF_m some finite group.

(Serre 53, see Ravenel 86, Chapter I, Lemma 1.1.8)

Remark

The non-torsion element in π 4k1(S 2k)\pi_{4k-1}(S^{2k}) may be seen explicitly in the minimal Sullivan model for the rational 2k-sphere, see there for more.

References

The original proof is due to

The statement is reviewed in

An entirely different proof, using only elementary concepts, is given in

  • S. S. Podkorytov, An Alternative Proof of a Weak Form of Serre’s Theorem, Journal of Mathematical Sciences July 2002, Volume 110, Issue 4, pp 2875–2881 (doi:10.1023/A:1015370800473)

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