Hatcher-Algebraic-Topology · Problem 4.1.21

Homotopy Theory

Mathematics — Graph Theory · postnikov-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.

Intuitive explanation

The key insight is that both constructions—truncating a Postnikov tower and then killing low homotopy groups, versus killing low groups first and then truncating—produce the same space up to homotopy equivalence because they're extracting the same "middle slice" of homotopy information. Since Postnikov stages kill high homotopy groups and $(m-1)$ -connected covers kill low ones, applying them in either order isolates exactly the homotopy groups in the range $[m,n]$ , making the final result independent of the order of operations. This commutativity justifies the unambiguous notation $X^n_m$ and explains why its homotopy groups are zero outside $[m,n]$ .

Intuitive explanation

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