Algebraic Topology · Problem 4F.3

Homotopy Theory — Allen Hatcher

Mathematics — Topology · spectra-homology-theories

Problem

Show that for any sequence Z1→Z2→⋯Z_1 \to Z_2 \to \cdots, the natural map lim→⁡ΩZn→Ωlim→⁡Zn\varinjlim \Omega Z_n \to \Omega \varinjlim Z_n is a weak homotopy equivalence, where the direct limits mean mapping telescopes.

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