Hatcher-Algebraic-Topology · Problem 3.1.13

Cohomology

Mathematics — Graph Theory · k-g-1-homotopy-classes-bijection

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

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

via the universal coefficient theorem.