Hatcher-Algebraic-Topology · Problem 2.2.3

Homology

Mathematics — Graph Theory · degree-theory

Problem

Let f:Sn→Snf: S^n \to S^n be a map of degree zero. Show that there exist points x,y∈Snx, y \in S^n with f(x)=xf(x) = x and f(y)=−yf(y) = -y. Use this to show that if FF is a continuous vector field defined on the unit ball DnD^n in Rn\mathbb{R}^n such that F(x)≠0F(x) \neq 0 for all xx, then there exists a point on ∂Dn\partial D^n where FF points radially outward and another point on ∂Dn\partial D^n where FF points radially inward.

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