AlgTop13: More applications of winding numbers

We define the degree of a function from the circle to the circle, and use that to show that there is no retraction from the disk to the circle, the Brouwer fixed point theorem, and a Lemma of Borsuk.

This is the 13th lecture of this beginner's course in Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW.

Category:

Education

Tags:

• AlgTop13
• Algebraic
• topology
• degree
• function
• circle
• retraction
• Brouwer
• fixed
• point
• theorem
• Lemma
• Borsuk
• Wildberger
• UNSW

[akash007ssp]

Hello Professor. I had a question. At around 3:56, you mention that "if we vary the function continuously, this degree will not change" What if I pick a new function g(alpha) = -f(alpha). f(alpha) being continuous, so is g(alpha) and I am unable to see why this does not affect the winding number/degree. Though I strongly feel this change is not continuous, I would like to have some more reasons and a little more formal justification for the above claim. Thanks for your time. Best -Akash

[UNSWelearning]

Hi akash007ssp The point is that we are considering only changing the function f continuously. In other words we must gradually go from f to another function without any abrupt disruption. It is not possible to go continuously from the function f(alpha) to g(alpha)=-f(alpha).

[DragonBuddah]

p1 is not a map from S1 to S1, it is rather a map from a point (the origin) to S1. A point is not topologically equivalent to S1.

[ja524309]

You don't explain why r(x) is continuous (and thus a retraction) during your proof of Brouwer's Fixed Point Theorem. It is pretty obvious, but nevertheless you should be a little more formal in your demonstration.

[charlesrkiss]

I guess I'm just questioning the generality of continuity; probably would be a good idea if there were some way around it! No pun intended, of course.

**ughhhh**

[charlesrkiss]

If there were "holes" everywhere; ie. "some real numbers didn't exist" (say, in reality =>discrete), everything would be multiply-connected and could be mapped from S3 to S2 to S1, etc.

Perhaps fractional exponents could fit more nicely into transformations between functions of different degrees: something that really bothers me alot(!) since since y=x^2 looks a lot like y=x^3/2. **ughhhh**

[charlesrkiss]

I dislike how in projective geometry a pole is mapped to infinity [AlgTop4: 3:06], ie. by either continuous and discrete approximations; and here [11:36], a pole (P(1), can't be mapped to degree 0.

This seems contradictory.

If an origin didn't exist, but was a "singularly missing point", D wouldn't be a disk and could be mapped to S1. And if the sphere were missing two antipodal points, it could be mapped to the disk, and so on.

Fixed points not being found, only supports the argument.