nLab second stable homotopy group of spheres

Idea

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

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

Related concepts

• first stable homotopy group of spheres

• third stable homotopy group of spheres

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

• Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Communication II, Doklady Akademii Nauk SSSR 19(5) (1938), 361-363

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

• Lev Pontryagin, Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 (pdf)

• George Whitehead, The $(n+2)$nd Homotopy Group of the $n$-Sphere, Annals of Mathematics Second Series, Vol. 52, No. 2 (Sep., 1950), pp. 245-247 (jstor:1969466)

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

• Lev Pontrjagin, Section 15 of: Smooth manifolds and their applications in homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akademii Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001)

These historical references are also listed, with brief commentary, in the first part of:

• Douglas Ravenel, Early work on the stable 2-stem

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)

See also:

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