Degree of a Map: Finding Antipodal Points for Odd Degree Functions

[kbfrob]

Homework Statement

Two questions: 1) Show that deg(f(g(x)) = deg(f)*deg(g)
2) f: Sⁿ -> Sⁿ
deg(f) is odd
then show there exists a pair of antipodal points that are mapped to antipodal points

The Attempt at a Solution

1) I have tried the method of just counting preimages, but i don't think this is the right direction. i know intuitively the deg of map between spheres is how many times you "wrap" the sphere around, and what i trying to show supports this, but i don't know how to show it rigorously.

2) We are dealing with antipodal maps in class, so i am trying to use them, but i have no idea how to use the fact that the degree of f is odd. i don't recall there being any fundamental difference between a map of odd and even degree.

 

[Hurkyl]

1) I have tried the method of just counting preimages, but i don't think this is the right direction.

What doesn't work about it?

 

[kbfrob]

it doesn't take into account the sign of the dervative at each of the preimages. in other words, the degree is not just the number of preimages. I'm not sure how to reconcile this.

 

[Hurkyl]

it doesn't take into account the sign of the dervative at each of the preimages. in other words, the degree is not just the number of preimages. I'm not sure how to reconcile this.

Well, what if you count preimages together with their sign? Or modify the maps so that they have positive derivative at each preimage?

 

[kbfrob]

alright i got it. thank you very much.

any ideas on the second question?
i don't even know how to start.

 

[Hurkyl]

alright i got it. thank you very much.

any ideas on the second question?
i don't even know how to start.

Nothing seems obvious to me. I suspect if I knew the content of the current chapter / previous couple chapters, something would leap out at me!

My best ideas at the moment are:
1. Try and apply some sort of fixed point theorem
2. Say something clever involving the quotient map Sⁿ --> Sⁿ / ~, where ~ is the relation that identifies antipodal points
3. Study some gadget that measures how different f(x) and f(-x) are from being antipodal (using -x to denote the point antipodal to x)

It's easy to see that odd degree is a necessary condition -- consider the standard maps $f(\theta) = k \theta$ of degree k on the circle.

 

[HallsofIvy]

If f is a function of odd degree, mapping Sⁿ to Sⁿ, A(p) be the function that maps each point to its antipodal point and look at f(p)- f(A(p)).