Algebraic Topology · Problem 3.1.13

Cohomology — Allen Hatcher

Mathematics — Topology · maps-to-eilenberg-maclane-space

Problem

Let

\langle X, Y \rangle

denote the set of basepoint-preserving homotopy classes of basepoint-preserving maps

X \to Y

. Using Proposition 1B.9, show that if

X

is a connected CW complex and

G

is an abelian group, then the map

\langle X, K(G, 1) \rangle \to H^1(X; G)

sending a map

f: X \to K(G, 1)

to the induced homomorphism

f_*: H_1(X) \to H_1(K(G, 1)) \approx G

is a bijection, where we identify

H^1(X; G)

with

\text{Hom}(H_1(X), G)

via the universal coefficient theorem.