Algebraic Topology · Problem 4.1.21

Homotopy Theory — Allen Hatcher

Mathematics — Topology · postnikov-towers-whitehead-towers

Problem

For this problem it is convenient to use the notations

X^n

for the

n^{th}

stage in a Postnikov tower for

X

and

X_m

for an

(m-1)

-connected covering of

X

, where

X

is a connected CW complex. Show that

(X^n)_m \simeq (X_m)^n

, so the notation

X^n_m

is unambiguous. Thus

\pi_i(X^n_m) \approx \pi_i(X)

for

m \leq i \leq n

and all other homotopy groups of

X^n_m

are zero.