nLab
degree of a continuous function

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

Given a continuous function between two connected closed oriented topological manifolds of the same dimension, its degree is a measure for how often the function “wraps its domain around its codomain”.

Definition

For XX is a connected closed oriented manifold of dimension nn, its top homology group H n(X)=H n(X;)H_n(X) = H_n(X; \mathbb{Z}) is isomorphic to \mathbb{Z}, where the generator 11 \in \mathbb{Z} is identified with the orientation class [ω X][\omega_X] of XX, the fundamental class of XX.

Definition

Given a continuous map f:XYf \colon X \to Y between two such manifolds, the homomorphism f *=H n(f):H n(X)H n(Y)f_\ast = H_n(f) \colon H_n(X) \to H_n(Y) is therefore specified by the integer nn such that f *[ω X]=n[ω Y]f_\ast [\omega_X] = n [\omega_Y]. This integer is called the degree of ff.

Computing the degree

We suppose throughout that XX and YY are connected closed oriented manifolds of the same dimension nn. The degree of a continuous function g:XYg \colon X \to Y is frequently computed according to the following considerations:

Properties

Hopf degree theorem

Proposition

(Hopf degree theorem)

Let nn \in \mathbb{N} be a natural number and XMfdX \in Mfd be a connected orientable closed manifold of dimension nn. Then the nnth cohomotopy classes [XcS n]π n(X)\left[X \overset{c}{\to} S^n\right] \in \pi^n(X) of XX are in bijection to the degree deg(c)deg(c) \in \mathbb{Z} of the representing functions, hence the canonical function

π n(X)S nK(,n)H n(X,) \pi^n(X) \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow} H^n(X,\mathbb{Z}) \;\simeq\; \mathbb{Z}

from nnth cohomotopy to nnth integral cohomology is a bijection.

(e.g. Kosinski 93, IX (5.8), Kobin 16, 7.5)

Poincaré–Hopf theorem

See at Poincaré–Hopf theorem.

Generalization to the Adams d-invariant

The Hopf degree of a map is a special case of its Adams d-invariant; see there for more.

Examples

References

Texbook accounts:

See also: