Algebraic Topology · Problem 4J.1

Homotopy Theory — Allen Hatcher

Mathematics — Topology · loopspace-of-a-suspension

Problem

Show that

\Omega\Sigma X

for a nonconnected CW complex

X

reduces to the connected case by showing that each path-component of

\Omega\Sigma X

is homotopy equivalent to

\Omega\Sigma\bigvee_\alpha X_\alpha

where the

X_\alpha

's are the components of

X