Algebraic Topology · Problem 4.1.23

Homotopy Theory — Allen Hatcher

Mathematics — Topology · mapping-cylinder-cw-pair-homotopy-type

Problem

f: X \to Y

is a map with

X

and

Y

homotopy equivalent to CW complexes, show that the pair

(M_f, X)

is homotopy equivalent to a CW pair, where

M_f

is the mapping cylinder. Deduce that the mapping cone

C_f

has the homotopy type of a CW complex.