nLab Classification of continuous maps of a complex into a sphere

This entry is about the articles

• Lev Pontrjagin:

  Classification of continuous maps of a complex into a sphere, Communication I

  Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

• Lev Pontrjagin:

  Classification of continuous maps of a complex into a sphere,

  Communication II

  Doklady Akademii Nauk SSSR 19(5) (1938), 361-363

  (this article contains a famous mistake, see also p. 6 of Hopkins, Singer‘s Quadratic Functions in Geometry, Topology, and M-Theory and Hopkins’s talk at Atiyah’s 80th Birthday conference, slide 8, 9:45)

• 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)

  (this article fixes the mistake)

(all three of which are available in English translation in Gamkrelidze 86)

or rather about their joint comprehensive exposition in

• Lev Pontrjagin:

  Smooth manifolds and their applications in homotopy theory

  Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955

  AMS Translation Series 2, Vol. 11, 1959

  doi:10.1142/9789812772107_0001

  pdf)

on differential topology, establishing the Pontryagin isomorphism between unstable Cohomotopy and cobordism classes of normally framed submanifolds, and applying it to the computation of the lowest couple of stable homotopy groups of spheres (the first and the second).

Together with

• René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, pdf),

due to which Pontryagin’s construction came to be mainly known as the Pontryagin-Thom construction, this lays the foundations of cobordism theory as such and as a tool in stable homotopy theory.

Chapter I – Smooth manifolds and smooth maps

1. Smooth manifolds

• differentiable manifold, differentiable map

• smooth manifold, smooth map

2. Embedding of smooth manifolds in Euclidean space

• submanifold

• Whitney embedding theorem

3. Improper points of smooth maps

• regular point, critical point

4. Non-degenerate singular points of smooth maps

Chapter II – Normally-framed manifolds

5. Smooth approximations to continuous maps and deformations

6. The basic method

• normally framed submanifold

• cobordism, cobordism class

• continuous homotopy implies smooth homotopy

• Pontryagin's theorem

7. Homology groups of framed manifolds

8. The suspension operation

Chapter III – The Hopf invariant

9. The homotopy classification of maps of $n$-dimensional manifolds into the $n$-sphere

• Hopf degree theorem

10. The Hopf invariant of maps of $\Sigma^{2k+1}$ into $S^{k+1}$

11. Framed manifolds with zero Hopf invariant

Chapter IV – Classification of maps between spheres

• homotopy groups of spheres

12. The rotation group of Euclidean space

• orthogonal group

13. Classification of maps to the 3-sphere into the 2-sphere

• Hopf fibration

14. Classification of maps of the $(n+1)$-sphere into the $n$-sphere

• first stable homotopy group of spheres

15. Classification of maps of the $(n+2)$-sphere into the $n$-sphere

• second stable homotopy group of spheres

category: reference