Algebraic Topology · Problem 21

Homotopy Theory — Allen Hatcher

Mathematics — Topology · product-kgn-homotopy-groups

Problem

Given a sequence of CW complexes

K(G_n, n)

n = 1, 2, \cdots

, let

X_n

be the CW complex formed by the product of the first

n

of these

K(G_n, n)

's. Via the inclusions

X_{n-1} \subset X_n

coming from regarding

X_{n-1}

as the subcomplex of

X_n

with

n^{th}

coordinate equal to a basepoint 0-cell of

K(G_n, n)

, we can then form the union of all the

X_n

's, a CW complex

X

. Show

\pi_n(X) \approx G_n

for all

n