nLab
first stable homotopy group of spheres

Contents

Context

Stable Homotopy theory

Cobordism theory

Contents

Idea

The first stable homotopy group of spheres (the first stable stem) is the cyclic group of order 2:

(1)π 1 s /2 [h ] [1] \array{ \pi_1^s &\simeq& \mathbb{Z}/2 \\ [h_{\mathbb{C}}] &\leftrightarrow& [1] }

where the generator [1]/2[1] \in \mathbb{Z}/2 is represented by the complex Hopf fibration S 3h S 2S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2.

from SS21

Properties

As the first framed bordism group

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring Ω fr\Omega^{fr}_\bullet of stably framed manifolds (see at MFr), the generator (1) is represented by the 1-sphere (with its left-invariant framing induced from the identification with the Lie group U(1))

π 1 s Ω 1 fr [h ] [S fr=1 1]. \array{ \pi_1^s & \simeq & \Omega_1^{fr} \\ [h_{\mathbb{C}}] & \leftrightarrow & [S^1_{fr=1}] \,. }

Moreover, the relation 2[S Lie 1]02 \cdot [S^1_{Lie}] \,\simeq\, 0 is represented by the bordism which is the complement of 2 open balls inside the 2-sphere.

References

The original computation via Pontryagin's theorem in cobordism theory:

with a more comprehensive account in:

Review:

Last revised on February 4, 2021 at 06:23:05. See the history of this page for a list of all contributions to it.