Let \(X\) consist of two copies of the solid torus \(\DD^2 \times S^1\), glued together by the identity map along the boundary torus \(S^1 \times S^1\). Compute the homology groups of \(X\).
Solution
Goal: Let \(U_1 = S^1 \times B^2\) and \(U_2 = S^1 \times B^2\) be two solid tori, glued along their boundaries \(\partial(S^1 \times B^2) = S^1 \times S^1 = T^2\) via the identity map \(\operatorname{id}_{\partial}\). Compute the integral homology groups \(H_*(M; \mathbb{Z})\) of the resulting closed 3-manifold \(M = S^1 \times S^2\).
1. Topological identification of \(M\).
1.1. \(B^2\) is the 2-disk \(D^2\), whose boundary is \(S^1\).
1.2. Gluing two copies of the 2-disk \(D^2\) along their common boundary via the identity map yields the 2-sphere \(S^2\): \[D^2 \cup_{\operatorname{id}_{S^1}} D^2 \cong S^2.\]
1.3. Since the gluing map on \(S^1 \times \partial B^2\) is \(\operatorname{id}_{S^1 \times \partial B^2} = \operatorname{id}_{S^1} \times \operatorname{id}_{S^1}\), the product structure factors through the gluing: \[M = (S^1 \times B^2) \cup_{\operatorname{id}} (S^1 \times B^2) \cong S^1 \times (B^2 \cup_{\operatorname{id}} B^2) \cong S^1 \times S^2.\]
1.4. The gluing map is the identity on both the \(S^1\) factor and the boundary \(S^1\) of \(B^2\), so the pushout distributes over the product: \((S^1 \times B^2) \cup_{\operatorname{id}} (S^1 \times B^2) \cong S^1 \times (B^2 \cup_{\operatorname{id}} B^2) \cong S^1 \times S^2\).
2. Compute \(H_*(S^1 \times S^2; \mathbb{Z})\) using the Künneth formula.
2.1. The homology groups of the circle \(S^1\) and the sphere \(S^2\) are:
\(H_0(S^1) \cong \mathbb{Z}\), \(H_1(S^1) \cong \mathbb{Z}\), \(H_k(S^1) = 0\) for \(k \ge 2\).
\(H_0(S^2) \cong \mathbb{Z}\), \(H_2(S^2) \cong \mathbb{Z}\), \(H_k(S^2) = 0\) for \(k \neq 0, 2\). <2>2. Since all homology groups of \(S^1\) and \(S^2\) are free abelian groups, all Tor terms \(\operatorname{Tor}_1^\mathbb{Z}(H_i(S^1), H_j(S^2))\) vanish. <2>3. The Künneth formula gives an isomorphism \(H_k(M) \cong \bigoplus_{i+j=k} (H_i(S^1) \otimes_\mathbb{Z} H_j(S^2))\) for each \(k \ge 0\):
\(k = 0\): \(H_0(S^1) \otimes H_0(S^2) = \mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z}\).
\(k = 1\): \((H_1(S^1) \otimes H_0(S^2)) \oplus (H_0(S^1) \otimes H_1(S^2)) = (\mathbb{Z} \otimes \mathbb{Z}) \oplus 0 \cong \mathbb{Z}\).
\(k = 2\): \((H_0(S^1) \otimes H_2(S^2)) \oplus (H_1(S^1) \otimes H_1(S^2)) \oplus (H_2(S^1) \otimes H_0(S^2)) = (\mathbb{Z} \otimes \mathbb{Z}) \oplus 0 \oplus 0 \cong \mathbb{Z}\).
\(k = 3\): \((H_1(S^1) \otimes H_2(S^2)) = \mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z}\).
\(k \ge 4\): All terms are 0, so \(H_k(M) = 0\).
2.4. Since \(H_i(S^1)\) and \(H_j(S^2)\) are free abelian, all Tor terms vanish and the Künneth formula reduces to the direct sum of tensor products; evaluating these for each \(k\) gives the groups in
2.3.
3. Alternative verification via the Mayer-Vietoris sequence.
3.1. Let \(A = S^1 \times B^2\) and \(B = S^1 \times B^2\).
3.2. \(A \simeq S^1\), \(B \simeq S^1\), and \(A \cap B \cong S^1 \times S^1 = T^2\).
3.3. The Mayer-Vietoris long exact sequence is: \[0 \to H_3(M) \xrightarrow{\partial} H_2(T^2) \xrightarrow{(i_*, j_*)} H_2(A) \oplus H_2(B) \to H_2(M) \xrightarrow{\partial} H_1(T^2) \xrightarrow{(i_*, j_*)} H_1(A) \oplus H_1(B) \to H_1(M) \to \widetilde{H}_0(T^2) \to 0.\]
3.4. \(H_2(A) = H_2(B) = 0\), so \(0 \to H_3(M) \xrightarrow{\partial} H_2(T^2) \cong \mathbb{Z} \to 0 \implies H_3(M) \cong \mathbb{Z}\).
3.5. The map \((i_*, j_*) \colon H_1(T^2) \cong \mathbb{Z}^2 \to H_1(A) \oplus H_1(B) \cong \mathbb{Z}^2\) maps the longitudinal circle generator \((1, 0) \mapsto (1, 1)\) and the meridian circle generator \((0, 1) \mapsto (0, 0)\).
Thus \(\ker(i_*, j_*) \cong \mathbb{Z}\) (generated by the meridian \((0, 1)\)) and \(\operatorname{coker}(i_*, j_*) \cong \mathbb{Z}\).
From \(0 \to H_2(M) \xrightarrow{\partial} \ker(i_*, j_*) \to 0\), we get \(H_2(M) \cong \mathbb{Z}\).
From \(0 \to \operatorname{coker}(i_*, j_*) \to H_1(M) \to 0\), we get \(H_1(M) \cong \mathbb{Z}\).
3.6. The Mayer–Vietoris sequence, together with the vanishing of \(H_2(A)\) and \(H_2(B)\) and the explicit form of \((i_*, j_*)\) on \(H_1(T^2)\), gives the short exact sequences that determine \(H_3(M)\), \(H_2(M)\), and \(H_1(M)\).
4. Conclusion.
4.1. The homology groups of \(M\) are: \[H_0(M; \mathbb{Z}) \cong \mathbb{Z}, \quad H_1(M; \mathbb{Z}) \cong \mathbb{Z}, \quad H_2(M; \mathbb{Z}) \cong \mathbb{Z}, \quad H_3(M; \mathbb{Z}) \cong \mathbb{Z},\] and \(H_k(M; \mathbb{Z}) = 0\) for all \(k \ge 4\).
4.2. The Künneth computation in 2 and the Mayer–Vietoris computation in 3 agree, giving the stated homology groups.