Topological degree

Degree of a circle endomorphism

Let be continuous. Then there is a unique continuous with the property and , #m/thm/homotopy so the following diagram commutes1

\begin{tikzcd} {[0,1]} && {\mathbb R} \\ \\ {\mathbb S^1} && {\mathbb S^1} \arrow["{\mathrm{ex}}"', from=1-1, to=3-1] \arrow["{f(1)\mathrm{ex}}", from=1-3, to=3-3] \arrow["f"{description}, from=3-1, to=3-3] \arrow["{f\mathrm{ex}}"{description}, from=1-1, to=3-3] \arrow["\varphi", dashed, from=1-1, to=1-3] \end{tikzcd}

Then the degree of is given by

which is always a whole number.

Proof

Without loss of generality we may assume , since otherwise we may use First we will show that if such a exists it is necessarily unique. Let with and . Then

for all , which may be the case iff for all . Since and are continuous so is , and thus is a constant map. Thus for all , i.e. .

Since is continuous it is uniformly continuous by the Heine-Cantor theorem, we can divide by with finite so that

for all integers . We write to denote the value of corresponding to some . whence it follows that and are not antipodes, namely

and therefore the Main branch of the complex logarithm is well-defined. We define as follows

which is continuous by properties of the Main branch of the complex logarithm. Additionally, and clearly .

All that's left to show is that . This is true since by definition and hence .

Generalisation to closed path

If is a closed continuous path, then we may define the winding number of around as

where

Ring isomorphism

is a ring with function multiplication as addition and composition as multiplication. Then is a ring isomorphism, since Circle endomorphisms are homotopic iff they are of equal degree and for all ,

Examples

Properties

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2010, Algebraische Topologie, pp. 37–41 ↩