Lemma 13.4.
For \(n \geq 2\text{,}\) the group \(\pi_n\bigl(\bigvee_\alpha S^n_\alpha\bigr)\) is free abelian, with basis the classes of the inclusions of the wedge summands.
Section 13.1 Technical Lemmas
Proof.
Suppose first that there are only finitely many summands. Give \(S^n\) the CW structure with one \(0\)-cell and one \(n\)-cell. Then \(\prod_\alpha S^n_\alpha\) has the product CW structure, with cells only in dimension \(0,n,2n,\ldots,|\alpha|n\text{.}\) Its \(n\)-skeleton is exactly \(\bigvee_\alpha S^n_\alpha\text{.}\) Since all other cells have dimension at least \(2n\text{,}\) the pair \(\bigl(\prod_\alpha S^n_\alpha, \bigvee_\alpha S^n_\alpha\bigr)\) is \((2n-1)\)-connected by Lemma 9.2. Since \(n \geq 2\) we have \(2n - 1 \geq n + 1\text{,}\) so the long exact sequence of the pair and the product formula of Chapter 5 give
\begin{equation*}
\pi_n\Bigl(\bigvee_\alpha S^n_\alpha\Bigr) \cong \pi_n\Bigl(\prod_\alpha S^n_\alpha\Bigr) \cong \prod_\alpha \pi_n(S^n_\alpha) \cong \bigoplus_\alpha \mathbb{Z},
\end{equation*}
using Corollary 11.12. This handles the case of finitely many summands.
The infinite case reduces to this one. Let \(\Phi \colon \bigoplus_\alpha \pi_n(S^n_\alpha) \to \pi_n\bigl(\bigvee_\alpha S^n_\alpha\bigr)\) be induced by the inclusions. Any map \(f \colon S^n \to \bigvee_\alpha S^n_\alpha\) has compact image, which is contained in the wedge of finitely many summands. So \([f]\) is in the image of \(\Phi\) by the finite case, and \(\Phi\) is surjective. Similarly, a null-homotopy of \(f\) has compact image, and again by the finite case \(\Phi\) is injective.
Remark 13.5.
For \(n = 1\) the Seifert–Van Kampen theorem (Theorem 4.19) shows that \(\pi_1\bigl(\bigvee_\alpha S^1_\alpha\bigr)\) is the free group on the summands. As soon as there is more than one circle, this group is not abelian.
Lemma 13.6.
Let \((X, A)\) be a CW pair. If \((X, A)\) is \(r\)-connected (\(r \geq 1\)) and \(A\) is \(s\)-connected (\(s \geq 0\)), then the quotient map \(X \to X/A\) induces a map
\begin{equation*}
\pi_i(X, A) \to \pi_i(X/A)
\end{equation*}
which is an isomorphism for \(i \leq r + s\) and a surjection for \(i = r + s + 1\text{.}\)
Proof.
Let \(CA = (A \times I)/(A \times \{1\})\) be the cone on \(A\) (Exercise 7.15), with \(A\) included as \(a \mapsto [(a,0)]\text{,}\) and form the CW complex \(Y = X \cup_A CA\text{.}\) The cone \(CA\) is a contractible subcomplex of \(Y\text{,}\) so the quotient map \(q \colon Y \to Y/CA = X/A\) is a homotopy equivalence (Proposition 7.17). The quotient map also defines a map of pairs \(\bar q \colon (Y, CA) \to (X/A, *)\text{,}\) and we get a commutative diagram
The left triangle commutes because the composite \((X, A) \to (Y, CA) \xrightarrow{\bar q} (X/A, *)\) is the quotient map of \((X, A)\text{.}\) The right triangle commutes because the composite \((Y, *) \to (Y, CA) \xrightarrow{\bar q} (X/A, *)\) is \(q\text{.}\) The map \(j_*\) is an isomorphism by the long exact sequence of \((Y, CA)\text{,}\) since \(CA\) is contractible. Hence \(\bar q_*\) is an isomorphism, and it suffices to study the map \(\pi_i(X, A) \to \pi_i(Y, CA)\text{.}\)
We apply Theorem 11.6 to the CW triad \((Y; X, CA)\text{,}\) whose intersection is \(X \cap CA = A\text{.}\) The pair \((X, A)\) is \(r\)-connected by hypothesis, and \((CA, A)\) is \((s+1)\)-connected. Indeed, the cone \(CA\) is contractible, so the long exact sequence gives \(\pi_i(CA, A) \cong \pi_{i-1}(A)\text{,}\) which vanishes for \(i \leq s+1\) because \(A\) is \(s\)-connected. Excision then gives that \(\pi_i(X, A) \to \pi_i(Y, CA)\) is an isomorphism for \(i < r + s + 1\) and a surjection for \(i = r + s + 1\text{.}\)
Now let \(n \geq 2\) and suppose we are given basepoint-preserving maps \(\varphi_\beta \colon S^n_\beta \to \bigvee_\alpha S^n_\alpha\text{,}\) for \(\alpha \in I\) and \(\beta \in J\text{.}\) Write \(W = \bigvee_\alpha S^n_\alpha\) and construct a space \(X\) as the pushout
That is, \(X = W \cup_{\varphi_\beta} \bigcup_\beta e^{n+1}_\beta\text{,}\) with characteristic maps \(\Phi_\beta\text{.}\) The next lemma identifies the homotopy groups of \(X\) up to degree \(n\text{.}\)
Lemma 13.7.
For \(X\) as above,
\begin{equation*}
\pi_i(X) =
\begin{cases}
\bigl(\bigoplus_{\alpha \in I} \mathbb{Z}\bigr) \,\big/\, \bigl\langle [\varphi_\beta] : \beta \in J \bigr\rangle & \text{if } i = n, \\[2pt]
0 & \text{if } i < n.
\end{cases}
\end{equation*}
Proof.
Cellular approximation (Corollary 9.3) gives \(\pi_i(X) = 0\) for \(i < n\text{.}\) For \(i = n\text{,}\) note that the pair \((X, W)\) is obtained by attaching \((n+1)\)-cells, so it is \(n\)-connected by Lemma 9.2. It fits into a long exact sequence
\begin{equation*}
\pi_{n+1}(X, W) \xrightarrow{\;\partial\;} \pi_n(W) \to \pi_n(X) \to \pi_n(X, W) = 0.
\end{equation*}
It follows that \(\pi_n(X) \cong \pi_n(W)/\operatorname{im}(\partial)\text{,}\) and \(\pi_n(W) \cong \bigoplus_\alpha \mathbb{Z}\) by Lemma 13.4.
We have \(X/W \cong \bigvee_\beta S^{n+1}_\beta\text{.}\) Since \(W\) is \((n-1)\)-connected and \(n + 1 \leq 2n - 1\text{,}\) Lemma 13.4, Lemma 13.6 show that
\begin{equation*}
\pi_{n+1}(X, W) \cong \pi_{n+1}\Bigl(\bigvee_\beta S^{n+1}_\beta\Bigr)
\end{equation*}
is free abelian, with basis the classes of the characteristic maps \(\Phi_\beta \colon (D^{n+1}, S^n) \to (X, W)\text{.}\) Since \(\partial[\Phi_\beta] = [\varphi_\beta]\text{,}\) we get \(\operatorname{im}(\partial) = \langle [\varphi_\beta] : \beta \in J \rangle\text{.}\)
Remark 13.8.
Lemma 13.7 says nothing about \(\pi_i(X)\) for \(i > n\text{;}\) in general these groups will be nonzero. Killing them is the work of the rest of the chapter.
