Algebraic Topology · Problem 38

Homotopy Theory — Allen Hatcher

Mathematics — Topology · pi3-s1-wedge-s2-not-finitely-generated

Problem

Show

\pi_3(S^1 \vee S^2)

is not finitely generated as a module over

\mathbb{Z}[\pi_1(S^1 \vee S^2)]

by considering Whitehead products in the universal cover, using the results in Example 4.52. Generalize this to

\pi_{i+j-1}(S^1 \vee S^i \vee S^j)

for

i, j > 1