Hatcher-Algebraic-Topology · Problem 17

Homotopy Theory

Mathematics — Graph Theory · eilenberg-maclane-uniqueness

Problem

Show that the map

\langle X, Y \rangle \to \operatorname{Hom}(\pi_n(X), \pi_n(Y))

[f] \mapsto f_*

, is a bijection if

X

is an

(n-1)

-connected CW complex and

Y

is a path-connected space with

\pi_i(Y) = 0

for

i > n

. Deduce that CW complex

K(G,n)

's are uniquely determined, up to homotopy type, by

G

and

n