Hatcher-Algebraic-Topology · Problem 4.F.2

Homotopy Theory

Mathematics — Graph Theory · spectra-and-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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