Algebraic Topology · Problem 3F.1

Cohomology — Allen Hatcher

Mathematics — Topology · reverse-double-mapping-telescope

Problem

Given maps

f_i: X_i \to X_{i+1}

for integers

i < 0

, show that the 'reverse mapping telescope' obtained by glueing together the mapping cylinders of the

f_i

's in the obvious way deformation retracts onto

X_0

. Similarly, if maps

f_i: X_i \to X_{i+1}

are given for all

i \in \mathbb{Z}

, show that the resulting 'double mapping telescope' deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the

f_i

's for

i

greater than a given number

n

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