The Winding Number

Definition and Notation

We write $\QTR{Large}{(S,A)}$ for a pair consisting of a topological space $\QTR{Large}{S}$ and subspace .
We write $\QTR{Large}{(T,B)}$ for a map $\QTR{Large}{T\ }$ such that
We write for the set of such maps.
Given $\QTR{Large}{(T,B)}$ we say that $\QTR{Large}{f}_{0}$ is relatively homotopic to if there is a homotopy $\QTR{Large}{T}$ from $\QTR{Large}{f}_{0}$ to $\QTR{Large}{f}_{1}$ such that . It is immediate that relatively homotopy is also an equivalence relation.

The Setting

We will be concerned with the Real Line the Circle $\QTR{Large}{S}^{1}$ , which we take to be and
We will also be concerned with the (continuous) map $\QTR{Large}{S}^{1}$ defined by the formula

The reason that we use the parameter " $\QTR{Large}{s}$ " rather than " $\QTR{Large}{t}$ " will become clear in the in the next lecture.
We will be concerned with maps such that Using the notation introduced above, .

Since there will be no ambiguity we use the notation $\QTR{Large}{[I,}$ as an abreviation for .
Finally, we will be concerned with maps such that and such that In particular,

This is the picture:

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Lemma:

Let and let $\QTR{Large}{y\in }$ $\QTR{Large}{S^{1}}$ .

such that.

(in fact we could set )
is a homeomorphism of open subintervals.

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Definition:

Suppose we are given a map such that and such that then, is called the winding number or degree of $\QTR{Large}{g.}$

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Lemma:

Let such that and such that then

Proof:

Let for all such that Let . By the usual limit argument . We want to show that $\QTR{Large}{l=1}$ .

Let . Since $\QTR{Large}{e}$ is single-valued (in fact a homeomorphism) on , for

But and thus if then there is

some with

We want to show that But follows from the fact that and

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Lemma:

Let then there exists a unique , such that and such that $\QTR{Large}{eg=f}$ . In particular we can define a winding number on $\QTR{Large}{[I,}$ by setting

Proof:

Start with and, using the fact that $\QTR{Large}{e}$ is a local homeomorphism define $\QTR{Large}{g}$ on the "largest possible" subinterval of $\QTR{Large}{I}$ .Now use a "lub" argument to show you can define it on all of $\QTR{Large}{I}$ .

Homework due May 2. Give a detailed proof of the lemma:

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Theorem:

The winding number is invariant under relative homotopy. that is if are relatively homotopic then

Proof(to follow)

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Consequences

The maps and the constant map , where are not relatively homotopic.

Proof: and
There does not exist a map (always read continuous) such that is the identity map.

Proof: $\QTR{Large}{e}$ and $\QTR{Large}{c}$ considered as maps in are relatively homotopic. The formula is

Here is the picture

composing this relative homotopy with $\QTR{Large}{r}$ would give a relative homotopy of $\QTR{Large}{e}$ and $\QTR{Large}{c}$ in $\QTR{Large}{[I,}$ , contradicting 1.
Every map such that is the identity map, is onto.

Proof: If not we could find a map contradicting 2.
Every map such that is the identity map, has a fixed point.

Proof: If not we could find a map contradicting 2.

Homework due May 2. prove 3. and 4. (hint the proofs are almost identical to the case.