Algebraic Topology · Problem 4.1.19

Homotopy Theory — Allen Hatcher

Mathematics — Topology · weak-homotopy-equivalence-common-cw-approximation

Problem

Consider the equivalence relation

\simeq_w

generated by weak homotopy equivalence:

X \simeq_w Y

if there are spaces

X = X_1, X_2, \cdots, X_n = Y

with weak homotopy equivalences

X_i \to X_{i+1}

X_i \leftarrow X_{i+1}

for each

i

. Show that

X \simeq_w Y

iff

X

and

Y

have a common CW approximation.