Hatcher-Algebraic-Topology · Problem 4.I.1

Homotopy Theory

Mathematics — Graph Theory · stable-splittings-of-spaces

Problem

If a connected CW complex

X

retracts onto a subcomplex

A

, show that

\Sigma X \simeq \Sigma A \vee \Sigma(X/A)

. [One approach: Show the map

\Sigma r + \Sigma q : \Sigma X \to \Sigma A \vee \Sigma(X/A)

induces an isomorphism on homology, where

r: X \to A

is the retraction and

q: X \to X/A

is the quotient map.]