Algebraic Topology · Problem 4F.2

Homotopy Theory — Allen Hatcher

Mathematics — Topology · spectra-homology-theories

Problem

For CW complexes XX and YY consider the suspension sequence ⟨X,Y⟩→Σ⟨ΣX,ΣY⟩→Σ⟨Σ2X,Σ2Y⟩→⋯\langle X,Y\rangle \xrightarrow{\Sigma} \langle \Sigma X, \Sigma Y\rangle \xrightarrow{\Sigma} \langle \Sigma^2 X, \Sigma^2 Y\rangle \to \cdots Show that if XX is a finite complex, these maps eventually become isomorphisms. [Use induction on the number of cells of XX and the five-lemma.]

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