nLab Classification of continuous maps of a complex into a sphere


This entry is about the articles

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

or rather about their joint comprehensive exposition in

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

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

2. Embedding of smooth manifolds in Euclidean space

3. Improper points of smooth maps

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

7. Homology groups of framed manifolds

8. The suspension operation

Chapter III – The Hopf invariant

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

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

11. Framed manifolds with zero Hopf invariant

Chapter IV – Classification of maps between spheres

12. The rotation group of Euclidean space

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

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

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

category: reference

Last revised on February 4, 2021 at 17:03:41. See the history of this page for a list of all contributions to it.