nLab Quelques propriétés globales des variétés différentiables

This entry is about the article

• René Thom:

  
  

  Quelques propriétés globales des variétés différentiables

  
  

  Comment. Math. Helv. 28, (1954). 17-86

  doi:10.1007/BF02566923

  dml:139072

  digiz:GDZPPN002056259

  pdf

on differential topology, proving Thom's theorem which identifies cobordism classes of normally oriented submanifolds with homotopy classes of maps into a universal Thom space $M SO(n)$.

Together with

• Lev Pontrjagin, Classification of continuous maps of a complex into a sphere, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

or rather its belated 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)

due to which Thom’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 – Properties of differentiable maps

1. Definitions

• critical point, regular point (p. 18)

3. Pre-image of a sub-manifold

• tubular neighbourhood (p. 21)

4. Pre-image of a sub-manifold under a t-regular map

• Thom's transversality theorem (p. 26)

Chapter II – Sub-manifolds and homology classes of a manifold

2. Complex associated to a closed subgroup of the orthogonal group

• Thom space (p. 29)

Chapter III – On a problem of Steenrod

Chapter IV – Cobordant differentiable manifolds

• bordism, cobordism (p. 65)

3.

4. Cobordant sub-manifolds

• normally oriented submanifold (p. 67)

• Pontryagin-Thom collapse (p. 69)

• cobordism classes of submanifolds (p. 71)

5. A fundamental theorem

• Thom's theorem (p. 74, 75)

category: reference