nLab first stable homotopy group of spheres

Contents

Context

Cobordism theory

Concepts of cobordism theory

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

relative bordism theories:

algebraic:

Contents

Idea

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

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

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

Properties

As the first framed bordism group

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring $\Omega^{fr}_\bullet$ 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))

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

Moreover, the relation $2 \cdot [S^1_{Lie}] \,\simeq\, 0$ 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.