Hatcher-Algebraic-Topology · Problem 3

Cohomology

Mathematics — Graph Theory · cup-product-projective-spaces-maps

Problem

(a) Using the cup product structure, show there is no map

\mathbb{R}P^n \to \mathbb{R}P^m

inducing a nontrivial map

H^1(\mathbb{R}P^m; \mathbb{Z}_2) \to H^1(\mathbb{R}P^n; \mathbb{Z}_2)

n > m

. What is the corresponding result for maps

\mathbb{C}P^n \to \mathbb{C}P^m

? (b) Prove the Borsuk--Ulam theorem by the following argument. Suppose on the contrary that

f: S^n \to \mathbb{R}^n

satisfies

f(x) \neq f(-x)

for all

x

. Then define

g: S^n \to S^{n-1}

g(x) = (f(x) - f(-x))/|f(x) - f(-x)|

, so

g(-x) = -g(x)

and

g

induces a map

\mathbb{R}P^n \to \mathbb{R}P^{n-1}

. Show that part (a) applies to this map.