Algebraic Topology · Problem 8

Cohomology — Allen Hatcher

Mathematics — Topology · cohomology-rings-different-coefficients

Problem

Let

X

\mathbb{C}P^2

with a cell

e^3

attached by a map

S^2 \to \mathbb{C}P^1 \subset \mathbb{C}P^2

of degree

p

, and let

Y = M(\mathbb{Z}_p, 2) \vee S^4

. Thus

X

and

Y

have the same 3-skeleton but differ in the way their 4-cells are attached. Show that

X

and

Y

have isomorphic cohomology rings with

\mathbb{Z}

coefficients but not with

\mathbb{Z}_p

coefficients.