Let \(f: S_\alpha^2 \vee S_\beta^2 \to S_\alpha^2 \vee S_\beta^2\) be the map which is the identity on the \(S_\alpha^2\) summand and which on the \(S_\beta^2\) summand is the sum of the identity map and a homeomorphism \(S_\beta^2 \to S_\alpha^2\). Let \(X\) be the mapping torus of \(f\). The mapping torus of the restriction of \(f\) to \(S_\alpha^2\) forms a subspace \(A = S^1 \times S_\alpha^2 \subset X\). Show that the maps \(\pi_2(A) \to \pi_2(X) \to \pi_2(X, A)\) form a short exact sequence \(0 \to \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to 0\), and compute the action of \(\pi_1(A)\) on these three groups. In particular, show the action of \(\pi_1(A)\) is trivial on \(\pi_2(A)\) and \(\pi_2(X, A)\) but is nontrivial on \(\pi_2(X)\).
Solution
Goal. For the mapping torus \(X = M_f\) of the self-map \(f\) of \(S^2_\alpha \vee S^2_\beta\), show \(\pi_2(A) \to \pi_2(X) \to \pi_2(X, A)\) is the short exact sequence \(0 \to \ZZ \to \ZZ \oplus \ZZ \to \ZZ \to 0\), and compute the \(\pi_1(A)\)-action on the three groups.
1. Identify the map \(f_*\) on \(\pi_2\).
1.1. \(\pi_2(S^2_\alpha \vee S^2_\beta) = \ZZ \oplus \ZZ\), generated by the classes \(\alpha\) and \(\beta\) of the two spheres.
the wedge of two \(2\)-spheres has \(\pi_2\) the free abelian group on the two sphere classes.
1.2. \(f_*(\alpha) = \alpha\) and \(f_*(\beta) = \beta + \alpha\).
\(f\) is the identity on \(S^2_\alpha\), and on \(S^2_\beta\) it is \(\id_\beta + h\) where \(h: S^2_\beta \to S^2_\alpha\) is a homeomorphism, so \(f_*(\beta) = \beta + h_*(\beta) = \beta \pm \alpha\); taking \(h\) of degree \(+1\), \(f_*(\beta) = \beta + \alpha\).
1.3. So \(f_* = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\) in the basis \((\alpha, \beta)\).
read off the columns from 2.2.
2. Compute \(\pi_2(X)\).
2.1. \(X = M_f\) fibers over \(S^1\) with fiber \(F = S^2_\alpha \vee S^2_\beta\).
the mapping torus is the total space of a fibration \(F \to M_f \to S^1\).
2.2. \(\pi_2(X) \cong \pi_2(F) = \ZZ \oplus \ZZ\).
in the long exact sequence of the fibration, \(\pi_2(S^1) = 0\) and \(\pi_3(S^1) = 0\), so \(\pi_2(F) \to \pi_2(X)\) is an isomorphism.
2.3. \(\pi_1(X) = \ZZ\).
\(\pi_1(F) = 0\) and \(\pi_0(F) = 0\), so \(\pi_1(X) \to \pi_1(S^1) = \ZZ\) is an isomorphism.
3. The subspace \(A\) and its homotopy groups.
3.1. \(A = S^1 \times S^2_\alpha\) is the mapping torus of \(f|_{S^2_\alpha} = \id\).
the restriction of \(f\) to \(S^2_\alpha\) is the identity, so its mapping torus is \(S^1 \times S^2_\alpha\).
3.2. \(\pi_2(A) = \pi_2(S^1 \times S^2) = \ZZ\) and \(\pi_1(A) = \ZZ\).
\(\pi_2(S^1 \times S^2) = \pi_2(S^2) = \ZZ\) and \(\pi_1(S^1 \times S^2) = \pi_1(S^1) = \ZZ\).
4. The short exact sequence.
4.1. The inclusion \(A \hookrightarrow X\) induces on \(\pi_2\) the map \(\ZZ \to \ZZ \oplus \ZZ\) sending the generator to \(\alpha\).
\(A = S^1 \times S^2_\alpha\) contributes the \(\alpha\) summand of \(\pi_2(X)\).
4.2. This map is injective, and its cokernel is \(\ZZ\).
the image is \(\ZZ\alpha \subset \ZZ\alpha \oplus \ZZ\beta\), and the quotient is \(\ZZ\beta \cong \ZZ\).
4.3. The boundary map \(\pi_2(X, A) \to \pi_1(A)\) is zero.
\(\pi_1(A) \to \pi_1(X)\) is an isomorphism (both \(\ZZ\)), so the boundary map in the LES of the pair is zero.
4.4. Hence \(0 \to \pi_2(A) \to \pi_2(X) \to \pi_2(X, A) \to 0\) is exact, i.e. \(0 \to \ZZ \to \ZZ \oplus \ZZ \to \ZZ \to 0\).
combine 4.1, 4.2, 4.3.
5. The \(\pi_1(A)\)-action.
5.1. The action on \(\pi_2(A) = \ZZ\) is trivial.
\(A = S^1 \times S^2\) is a product, and \(\pi_1(S^1)\) acts trivially on \(\pi_2(S^2)\) (the loop in the \(S^1\) factor does not move the \(S^2\) factor).
5.2. The action on \(\pi_2(X) = \ZZ \oplus \ZZ\) is nontrivial, given by \(f_*\).
the generator of \(\pi_1(X) = \ZZ\) acts on \(\pi_2(X)\) by the monodromy \(f_* = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\), which sends \(\beta \mapsto \beta + \alpha \neq \beta\).
5.3. The action on \(\pi_2(X, A) = \ZZ\) is trivial.
\(\pi_2(X, A) \cong \ZZ\beta\) (the quotient by \(\ZZ\alpha\)), and \(f_*\) acts trivially on the quotient \(\ZZ\beta\) (since \(f_*(\beta) = \beta + \alpha \equiv \beta \pmod{\ZZ\alpha}\)).
6. Q.E.D.
4.4 gives the short exact sequence; 5 gives the action (trivial on \(\pi_2(A)\) and \(\pi_2(X,A)\), nontrivial on \(\pi_2(X)\)).