A.9. BROUWER FIXED POINTS 483

Corollary A.9.3 Let Dn be the closed ball of radius 1 centered at 0 in Rn,n ≥ 1. Thenthere does not exist a function g : Dn→ Sn−1 which is continuous and leaves all points ofSn−1 unchanged.

Proof: Suppose there were such a map. Then letting i be the inclusion map of Sn−1 intoDn it follows that g◦ i = id on Sn−1 and also i◦g = id on Dn. Also for t ∈ [0,1] ,

t (id)(x)+(1− t)(i◦g)(x) ∈ Dn

t (id)(x)+(1− t)(g◦ i)(x) = x ∈ Sn−1

This is by convexity of Dn. Therefore, g ◦ i, i ◦ g are both homotopy inverses on Sn−1 andDn respectively, and so by Theorem A.3.9, it follows that Hn−1 (Dn) and Hn−1

(Sn−1

)are

isomorphic, but this is certainly not the case because the first is 0 orZ depending on whethern > 1 or n = 1. The second is Z if n > 1 and if n = 1, it is H0

(S0)= Z⊕Z because in the

last case, S0 has two path components. ■With this, it is easy to prove the Brouwer fixed point theorem.

Corollary A.9.4 Let Dn be the closed unit ball, n≥ 1 and let h : Dn→ Dn be continuous.Then h has a fixed point.

Proof: If h has no fixed point, consider the mapping g in the following picture whichwould deliver a retraction onto Sn−1 the boundary of Dn. ■

h(x)

x

g(x)

Definition A.9.5 If a set A has the property that whenever f : A→ A is continuous, thereis a fixed point, then we say that A has the fixed point property.

Note that if two sets are homeomorphic and one has the fixed point property, then sodoes the other. Letting f : A→ Â be a homeomorphism with A having the fixed pointproperty and letting g : Â→ Â be continuous, then f−1 ◦g◦ f : A→ A and is continuous soit has a fixed point x. Then g( f (x)) = f (x) and so g also has a fixed point.

Corollary A.9.6 If C is any compact convex subset of Rn for n ≥ 1, and if f : C→ C iscontinuous, then f has a fixed point.

Proof: Let P be the continuous projection map onto C. Then take B a large ball whichcontains C. Consider f ◦P : B→ B. It has a fixed point x and so f (P(x)) = x. Since fmaps to C, it follows that x ∈C and so P(x) = x. Hence f (x) = x. ■

An examination of the argument used shows the following.

Corollary A.9.7 Suppose K is a continuous retraction of C where C has the fixed pointproperty. Then so does K.