nLab second stable homotopy group of spheres



Stable Homotopy theory

Cobordism theory



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 }


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These historical references are also listed, with brief commentary, in the first part of: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


  • 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)

See also:

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

Last revised on June 23, 2023 at 10:39:42. See the history of this page for a list of all contributions to it.