nLab 2-sphere

Idea

The 2-sphere $S^2$ is the ordinary sphere of dimension $2$, hence the $n$-sphere for $n = 2$.

Equipped with its canonical complex structure this is known as the Riemann sphere.

Related concepts

• geometric quantization of the 2-sphere

• spherical braid group

Properties

Homotopy groups

Proposition

The 3rd homotopy group of the 2-sphere $S^2$ is freely generated by the complex Hopf fibration $h_{\mathbb{C}}$:

$$\pi_3\big(S^2\big) \simeq \mathbb{Z} \,.$$

This is essentially due to Hopf 1931 Satz I.

Proof

Once one knows that that

1. $\pi_{n \geq 2}(S^1) = 0$

2. the complex Hopf fibration is a circle principal bundle, hence a Serre fibration with fiber $S^1$,

this follows by exactness of the corresponding long exact sequence of homotopy groups:

$$\underset{0} {\underbrace{ \pi_3(S^1) }} \longrightarrow \underset{\mathbb{Z}} {\underbrace{ \pi_3(S^3) }} \overset{ (h_{\mathbb{C}})_\ast }{\longrightarrow} \pi_3(S^2) \longrightarrow \underset{0} {\underbrace{ \pi_2(S^1) \mathrlap{\,.} }}$$

Remark

Under stabilization the complex Hopf fibration hence represents the first stable homotopy group of spheres.

In fact:

Proposition

All higher homotopy groups of the 2-sphere are non-trivial: $\pi_{\geq 2}(S^2) \neq 0$.

(Ivanov, Mikhailov & Wu 2016).

References

General

• Wikipedia: Sphere

• Wikipedia: 2-sphere

Original discussion of $\pi_3(S^2)$ via the complex Hopf fibration:

• Heinz Hopf: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Mathematische Annalen 104 (1931) 637–665 [doi:10.1007/BF01457962]

Proof that all the higher homotopy groups of the 2-sphere are non-trivial:

• Sergei O. Ivanov, Roman Mikhailov, Jie Wu: On nontriviality of homotopy groups of spheres, Homology, Homotopy and Applications 18 2 (2016) [doi:10.4310/HHA.2016.v18.n2.a18, arXiv:1506.00952]

Loop space of the 2-sphere

On the loop space $\Omega S^2$ of the 2-sphere

cf. also

• Neil Strickland, section 18 of: A Bestiary of Topological Objects [pdf, pdf]

in relation to braid groups:

• Frederick R. Cohen, J. Wu: On Braid Groups, Free Groups, and the Loop Space of the 2-Sphere, in: Categorical Decomposition Techniques in Algebraic Topology, in Progress in Mathematics 215, Birkhäuser (2003) 93-105 [doi:10.1007/978-3-0348-7863-0_6]

and regarded as a classifying space, $\Omega S^2 \,\simeq\, B \Omega^2 S^2$, (for “$\mathbf{l}$ine” bundles in nonabelian cohomology):

• Jack Morava: A homotopy-theoretic context for CKM/Birkhoff renormalization [arXiv:2307.10148, spire:2678618]

• Jack Morava: Some very low-dimensional algebraic topology [arXiv:2411.15885]