nLab first stable homotopy group of spheres



Stable Homotopy theory

Cobordism theory



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


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.


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

with a more comprehensive account in:

See also:

  • Mehmet Kirdar, On the First, the Second and the Third Stems of the Stable Homotopy Groups of Spheres [arXiv:2107.06103]


Discussion in homotopy type theory:

Last revised on June 23, 2023 at 10:40:01. See the history of this page for a list of all contributions to it.