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Section 6.5 Examples and First Applications

Everything in this section is a consequence of Theorem 6.14, and the arguments are typical of how the long exact sequence is used in practice.

i.
\(\Rightarrow\) (ii): For \(n \geq 1\text{,}\) the segment
\begin{equation*} \pi_{n+1}(X, A, x_0) = 0 \xrightarrow{\;\partial\;} \pi_n(A, x_0) \xrightarrow{\;i_*\;} \pi_n(X, x_0) \xrightarrow{\;j_*\;} \pi_n(X, A, x_0) = 0 \end{equation*}
shows that \(\ker(i_*) = \operatorname{im}(\partial) = 0\) and \(\operatorname{im}(i_*) = \ker(j_*) = \pi_n(X, x_0)\text{.}\)
(ii) \(\Rightarrow\) (i): Let \(n \geq 1\) and consider
\begin{equation*} \pi_n(A, x_0) \xrightarrow{\;i_*\;} \pi_n(X, x_0) \xrightarrow{\;j_*\;} \pi_n(X, A, x_0) \xrightarrow{\;\partial\;} \pi_{n-1}(A, x_0) \xrightarrow{\;i_*\;} \pi_{n-1}(X, x_0). \end{equation*}
Since \(i_*\) is surjective on \(\pi_n\text{,}\) exactness at \(\pi_n(X, x_0)\) gives \(\ker(j_*) = \pi_n(X, x_0)\text{,}\) i.e. \(j_* = 0\text{.}\) Since \(i_*\) is injective on \(\pi_{n-1}\) (for \(n - 1 \geq 1\) by hypothesis, and for \(n - 1 = 0\) because both \(\pi_0(A)\) and \(\pi_0(X)\) are single points), exactness at \(\pi_{n-1}(A, x_0)\) gives \(\operatorname{im}(\partial) = 0\text{,}\) so \(\partial^{-1}(0) = \pi_n(X, A, x_0)\text{.}\) Exactness at \(\pi_n(X, A, x_0)\) then gives \(\pi_n(X, A, x_0) = \partial^{-1}(0) = \operatorname{im}(j_*) = 0\text{.}\)
For the final statement, recall from Chapter 5 that a homotopy equivalence induces isomorphisms on all homotopy groups, so (ii) holds.

Example 6.18. Contractible ambient space.

Suppose \(X\) is contractible, for instance \(X = D^n\text{,}\) or \(X = CA\) the cone on \(A\text{.}\) Then \(\pi_k(X, x_0) = 0\) for all \(k \geq 0\text{,}\) and the long exact sequence
\begin{equation*} 0 = \pi_k(X, x_0) \to \pi_k(X, A, x_0) \xrightarrow{\;\partial\;} \pi_{k-1}(A, x_0) \to \pi_{k-1}(X, x_0) = 0 \end{equation*}
shows that \(\partial \colon \pi_k(X, A, x_0) \to \pi_{k-1}(A, x_0)\) is an isomorphism for all \(k \geq 2\text{:}\) it is a homomorphism with \(\ker(\partial) = \operatorname{im}(j_*) = 0\) and \(\operatorname{im}(\partial) = \ker(i_*) = \pi_{k-1}(A, x_0)\text{.}\) In particular,
\begin{equation*} \pi_k(D^n, S^{n-1}) \;\cong\; \pi_{k-1}(S^{n-1}) \qquad\text{for all } k \geq 2,\; n \geq 2. \end{equation*}
For example, \(\pi_2(D^2, S^1) \cong \pi_1(S^1) \cong \mathbb{Z}\text{,}\) by Theorem 2.16. Geometrically, the isomorphism sends a disc in \(D^2\) with boundary on \(S^1\) to the winding number of its boundary circle.

Example 6.19. A nonabelian \(\pi_2(X, A)\).

Let \(X = D^2 \vee D^2\) be two discs glued at a boundary point, and let \(A = S^1 \vee S^1 \subseteq X\) be the union of their boundary circles, with \(x_0\) the wedge point. Then \(X\) is contractible, so by Example 6.18
\begin{equation*} \pi_2(X, A, x_0) \;\cong\; \pi_1(S^1 \vee S^1, x_0) \;\cong\; F_2, \end{equation*}
the free group on two generators, by the Seifert–Van Kampen theorem (Theorem 4.19). This group is not abelian, showing that the restriction \(n \geq 3\) in Proposition 6.7 cannot be removed.

Exercise 6.20.

Let \(r \colon X \to A\) be a retraction onto a subspace \(A \subseteq X\) (so \(r|_A = \mathrm{id}_A\)), and let \(x_0 \in A\text{.}\) Show that for every \(n \geq 2\) there is a split short exact sequence
\begin{equation*} 0 \to \pi_n(A, x_0) \xrightarrow{\;i_*\;} \pi_n(X, x_0) \xrightarrow{\;j_*\;} \pi_n(X, A, x_0) \to 0, \end{equation*}
and deduce that \(\pi_n(X, x_0) \cong \pi_n(A, x_0) \oplus \pi_n(X, A, x_0)\text{.}\) Conclude that \(\pi_2(X, A, x_0)\) is abelian whenever \(A\) is a retract of \(X\text{.}\)

Exercise 6.21.

Let \((X, A)\) be a pair with \(X\) simply connected and \(A\) path-connected. Show that \(\pi_1(X, A, x_0)\) consists of a single point. Give an example of a pair \((X, A)\) with \(X\) and \(A\) path-connected for which \(\pi_1(X, A, x_0)\) has more than one element.
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