# nLab Classification of continuous maps of a complex into a sphere

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:

This entry is about the articles

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

• Classification of continuous maps of a complex into a sphere,

Communication II

(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)

• Homotopy classification of mappings of an $(n+2)$-dimensional sphere on an $n$-dimensional one

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

or rather about their joint comprehensive exposition in

• 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

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.

# Contents

## Chapter IV – Classification of maps between spheres

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

category: reference

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