Week 7 homework, Math 446.
1. Prove Sperner's lemma for 4-colorings of the vertices of a subdivision
S of the 3-simplex (tetrahedron) T: if one colors the vertices of T
4 distinct
colors, and any vertex of S is colored one of the colors of the vertices
of the face of T in which it lies, then there will be one tetrahedron
in
S with 4 distinct colored vertices (see
http://www.cut-the-knot.com/Curriculum/Geometry/SpernerLemma.shtml
for a hint). Use this to prove Brouwer's fixed point
theorem for the 3-ball. Discuss generalizations to higher dimensions.
Can you show that the antipodal map -id: S^n -> S^n is not homotopic
to the identity using Brouwer's fixed point theorem for B^n?
2. A topological space T has the fixed point property if for any map
f: T->T, f has a fixed point, i.e. a point x in T such that f(x)=x.
Which of the following has the fixed point property?
a. (solid) square
b. circle
c. 2-sphere
d. R^2
e. T^2
f. The letter T (thought of as a graph).