nLab
second stable homotopy group of spheres

Contents

Context

Stable Homotopy theory

Cobordism theory

Contents

Idea

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

(1)π 1 s /2 \array{ \pi_1^s &\simeq& \mathbb{Z}/2 }

References

Pontryagin had announced the computation in:

  • Lev Pontrjagin, Sur les transformations des sphères en sphères (pdf) in: Comptes Rendus du Congrès International des Mathématiques – Oslo 1936 (pdf)

The explanation of the proof strategy via Pontryagin's theorem in cobordism theory appears in

but there was a mistake in the proof left, fixed in Pontryagin 50 and, independently and by different means, in Whitehead 50:

A more comprehensive account of the computation and the cobordism theory behind it (Pontryagin's theorem) was then given in:

These historical references are also listed, with brief commentary, in the first part of:

For more historical commentary see p. 6 of Hopkins&Singer‘s Quadratic Functions in Geometry, Topology, and M-Theory Hopkins’s talk at Atiyah’s 80th Birthday conference, slide 8, 9:45.

Review:

  • Guozhen Wang, Zhouli Xu, Section 2.5 of: A survey of computations of homotopy groups of Spheres and Cobordisms, 2010 (pdf)

  • Andrew Putman, Section 10 of: Homotopy groups of spheres and low-dimensional topology (pdf, pdf)

Last revised on February 26, 2021 at 11:50:50. See the history of this page for a list of all contributions to it.