Algebraic Topology · Problem 2.2.3

Homology — Allen Hatcher

Mathematics — Topology · degree-theory

Problem

Let

f: S^n \to S^n

be a map of degree zero. Show that there exist points

x, y \in S^n

with

f(x) = x

and

f(y) = -y

. Use this to show that if

F

is a continuous vector field defined on the unit ball

D^n

\mathbb{R}^n

such that

F(x) \neq 0

for all

x

, then there exists a point on

\partial D^n

where

F

points radially outward and another point on

\partial D^n

where

F

points radially inward.