Section 11.3 Freudenthal’s Suspension Theorem

Definition 11.8.

For a based space \(X\text{,}\) recall the (reduced) suspension

\begin{equation*} \Sigma X = (X \times I) \,/\, \bigl(X \times \{0\} \,\cup\, X \times \{1\} \,\cup\, \{x_0\} \times I\bigr) \;\cong\; S^1 \wedge X. \end{equation*}

The suspension homomorphism is the map

\begin{equation*} \Sigma \colon \pi_q(X) \to \pi_{q+1}(\Sigma X), \qquad [f] \mapsto [\mathrm{id}_{S^1} \wedge f], \end{equation*}

where we use \(S^{q+1} \cong S^1 \wedge S^q\text{.}\)

For \(q \geq 1\) the map \(\Sigma\) is a group homomorphism, and it is natural in \(X\) (Exercise 11.22). If \(X\) is a CW complex based at a \(0\)-cell, then \(\Sigma X\) is again a CW complex, and it is the union of two subcomplexes, the upper and lower (reduced) cones

\begin{equation*} C_+ X = \text{image of } X \times [\tfrac12, 1], \qquad C_- X = \text{image of } X \times [0, \tfrac12]. \end{equation*}

Both cones are contractible, and they meet in the image of \(X \times \{\tfrac12\}\text{,}\) a copy of \(X\) (Figure 11.9).

Figure 11.9. The suspension \(\Sigma X\) as the union of two cones along \(X\text{.}\) The picture shows the unreduced suspension. In the reduced suspension the segment \(\{x_0\} \times I\) is also collapsed.

Theorem 11.10. Freudenthal Suspension Theorem.

If \(X\) is an \(n\)-connected CW complex (based at a \(0\)-cell) with \(n \geq 0\text{,}\) then the suspension homomorphism

\begin{equation*} \Sigma \colon \pi_q(X) \to \pi_{q+1}(\Sigma X) \end{equation*}

is an isomorphism for \(q < 2n + 1\) and a surjection for \(q = 2n + 1\text{.}\) In particular, for \(n \geq 1\text{,}\)

\begin{equation*} \Sigma \colon \pi_q(S^n) \to \pi_{q+1}(S^{n+1}) \end{equation*}

is an isomorphism for \(q < 2n - 1\) and a surjection for \(q = 2n - 1\text{.}\)

Proof.

View \(\Sigma X\) as the union of the upper and lower cones \(C_+ X\) and \(C_- X\text{,}\) intersecting in a copy of \(X\text{.}\) Both cones are contractible, so the long exact sequences of the pairs \((C_+ X, X)\) and \((\Sigma X, C_- X)\) give natural isomorphisms

\begin{equation*} \partial \colon \pi_{q+1}(C_+ X, X) \xrightarrow{\;\cong\;} \pi_q(X), \qquad j_* \colon \pi_{q+1}(\Sigma X) \xrightarrow{\;\cong\;} \pi_{q+1}(\Sigma X, C_- X). \end{equation*}

Under these identifications, \(\Sigma\) agrees with the map \(i_*\) induced by the inclusion, that is, the square

commutes.

Since \(X\) is \(n\)-connected, both pairs \((C_+ X, X)\) and \((C_- X, X)\) are \((n+1)\)-connected (this follows from contractibility of the cones together with the long exact sequence in homotopy). Now apply Theorem 11.6, with both connectivities equal to \(n+1\text{,}\) to the CW triad \((\Sigma X; C_+ X, C_- X)\text{,}\) whose intersection \(X\) is connected. It shows that \(i_*\) is an isomorphism for \(q + 1 < 2(n+1)\text{,}\) i.e. \(q < 2n + 1\text{,}\) and a surjection for \(q + 1 = 2(n+1)\text{,}\) i.e. \(q = 2n + 1\text{.}\) The statement for spheres is the case \(X = S^n\text{,}\) which is \((n-1)\)-connected.

Remark 11.11.

Example 11.4 is exactly the case \(X = S^1 \vee S^1\) (so \(n = 0\)) of this proof. Indeed, \(\Sigma X = S^2 \vee S^2\text{,}\) and \(C_\pm X\) are the wedges of the northern and southern hemispheres. By Example 11.7,

\begin{equation*} \Sigma \colon \pi_1(S^1 \vee S^1) \to \pi_2(S^2 \vee S^2) \end{equation*}

is the abelianization \(F_2 \to \mathbb{Z}^2\text{.}\) So at the edge \(q = 2n + 1\) of the range, \(\Sigma\) is surjective but in general not injective. Example 11.13 gives an example with spheres.

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