For CW complexes \(X\) and \(Y\) consider the suspension sequence
\[\langle X, Y \rangle \stackrel{\Sigma}{\longrightarrow} \langle \Sigma X, \Sigma Y \rangle \stackrel{\Sigma}{\longrightarrow} \langle \Sigma^2 X, \Sigma^2 Y \rangle \longrightarrow \cdots\]
Show that if \(X\) is a finite complex, these maps eventually become isomorphisms.
Solution
We prove stabilization by induction on the number of cells of the finite CW complex \(X\).
For \(X=S^q\), the suspension map is \[ \pi_{q+r}(\Sigma^rY) \longrightarrow \pi_{q+r+1}(\Sigma^{r+1}Y). \] For large \(r\), the space \(\Sigma^rY\) is highly connected: after suspending sufficiently often, its connectivity grows by one at each suspension. Freudenthal’s suspension theorem therefore implies that the displayed map is an isomorphism for all sufficiently large \(r\). The same is true for a finite wedge of spheres.
Now suppose \(X\) is obtained from a finite subcomplex \(A\) by attaching one \(q\)-cell. There is a cofibration \[ S^{q-1}\xrightarrow{\varphi}A\longrightarrow X\longrightarrow S^q \] and hence, after suspending \(r\) times and mapping into \(\Sigma^rY\), the associated Puppe sequence gives an exact sequence of homotopy classes. Suspending once more gives a second such exact sequence, and the suspension maps form a commutative diagram between them.
By the induction hypothesis, the suspension maps involving \(A\) are isomorphisms for all sufficiently large \(r\). By the sphere case, the maps involving \(S^{q-1}\) and \(S^q\) are also isomorphisms for sufficiently large \(r\). After increasing \(r\) if necessary, the relevant terms are groups, and the five-lemma applied to the two Puppe sequences shows that \[ \langle\Sigma^rX,\Sigma^rY\rangle \longrightarrow \langle\Sigma^{r+1}X,\Sigma^{r+1}Y\rangle \] is an isomorphism.
Since \(X\) has only finitely many cells, the induction terminates. Therefore \[ \boxed{\text{the suspension maps are isomorphisms for all sufficiently large }r.} \]