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

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

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These historical references are also listed, with brief commentary, in the first part of: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