Hatcher-Algebraic-Topology · Problem 14

Cohomology

Mathematics — Graph Theory · quotient-map-rp-cp-cohomology

Problem

Let

q: \mathbb{R}P^\infty \to \mathbb{C}P^\infty

be the natural quotient map obtained by regarding both spaces as quotients of

S^\infty

, modulo multiplication by real scalars in one case and complex scalars in the other. Show that the induced map

q^*: H^*(\mathbb{C}P^\infty; \mathbb{Z}) \to H^*(\mathbb{R}P^\infty; \mathbb{Z})

is surjective in even dimensions by showing first by a geometric argument that the restriction

q: \mathbb{R}P^2 \to \mathbb{C}P^1

induces a surjection on

H^2

and then appealing to cup product structures. Next, form a quotient space

X

\mathbb{R}P^\infty \sqcup \mathbb{C}P^n

by identifying each point

x \in \mathbb{R}P^{2n}

with

q(x) \in \mathbb{C}P^n

. Show there are ring isomorphisms

H^*(X; \mathbb{Z}) \approx \mathbb{Z}[\alpha]/(2\alpha^{n+1})

and

H^*(X; \mathbb{Z}_2) \approx \mathbb{Z}_2[\alpha, \beta]/(\beta^2 - \alpha^{2n+1})

, where

|\alpha| = 2

and

|\beta| = 2n+1

. Make a similar construction and analysis for the quotient map

q: \mathbb{C}P^\infty \to \mathbb{H}P^\infty