Algebraic Topology · Problem 2.2.12

Homology — Allen Hatcher

Mathematics — Topology · homology-induced-maps

Problem

Show that the quotient map

S^1 \times S^1 \to S^2

collapsing the subspace

S^1 \vee S^1

to a point is not nullhomotopic by showing that it induces an isomorphism on

H_2

. On the other hand, show via covering spaces that any map

S^2 \to S^1 \times S^1

is nullhomotopic.