Algebraic Topology · Problem 3.1.11

Cohomology — Allen Hatcher

Mathematics — Topology · moore-space-cohomology-splitting

Problem

Let

X

be a Moore space

M(\mathbb{Z}_m, n)

obtained from

S^n

by attaching a cell

e^{n+1}

by a map of degree

m

. (a) Show that the quotient map

X \to X/S^n = S^{n+1}

induces the trivial map on

\tilde{H}_i(-; \mathbb{Z})

for all

i

, but not on

H^{n+1}(-; \mathbb{Z})

. Deduce that the splitting in the universal coefficient theorem for cohomology cannot be natural. (b) Show that the inclusion

S^n \hookrightarrow X

induces the trivial map on

\tilde{H}^i(-; \mathbb{Z})

for all

i

, but not on

H_n(-; \mathbb{Z})