- Compute the homology of the Klein bottle using the Mayer-Vietoris sequence and a decomposition \(K = M \disjoint_f M\)
Solution
1. Decomposition of the Klein bottle into two Möbius bands:
1.1. The Klein bottle \(K\) is homeomorphic to the union of two Möbius bands \(M_1\) and \(M_2\) glued along their boundary circles:\[ K = M_1 \cup_\partial M_2. \]Slightly thicken \(M_1\) and \(M_2\) to open sets \(U, V \subset K\).
standard decomposition of the Klein bottle.
1.2. The homotopy types of the pieces are:- \(U \simeq S^1\) (core circle \(a\) of \(M_1\)), so \(H_0(U) \cong \mathbb{Z}\), \(H_1(U) \cong \mathbb{Z} \langle a \rangle\), and \(H_k(U) = 0\) for \(k \ge 2\).- \(V \simeq S^1\) (core circle \(b\) of \(M_2\)), so \(H_0(V) \cong \mathbb{Z}\), \(H_1(V) \cong \mathbb{Z} \langle b \rangle\), and \(H_k(V) = 0\) for \(k \ge 2\).- \(U \cap V \simeq S^1\) (the common boundary circle \(\gamma\)), so \(H_0(U \cap V) \cong \mathbb{Z}\), \(H_1(U \cap V) \cong \mathbb{Z} \langle \gamma \rangle\), and \(H_k(U \cap V) = 0\) for \(k \ge 2\).
deformation retract of Möbius band onto its core circle and collar neighborhood onto boundary.
2. Inclusion homomorphisms on homology:
2.1. The boundary circle of a Möbius band traverses its core circle twice:The inclusion \(i_1: U \cap V \to U\) induces \((i_1)_*: H_1(U \cap V) \to H_1(U)\) with \((i_1)_*([\gamma]) = 2[a]\).Similarly, the inclusion \(i_2: U \cap V \to V\) induces \((i_2)_*: H_1(U \cap V) \to H_1(V)\) with \((i_2)_*([\gamma]) = 2[b]\).
degree 2 wrapping of the boundary circle around the core of a Möbius band.
2.2. In the Mayer–Vietoris sequence, the map \(\Phi_1 = ((i_1)_*, -(i_2)_*): H_1(U \cap V) \to H_1(U) \oplus H_1(V)\) is given by:\[ \Phi_1: \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}, \qquad 1 \mapsto (2, -2). \]
Mayer–Vietoris homomorphism definition.
3. Computation of homology groups via Mayer–Vietoris:
3.1. The Mayer–Vietoris sequence is:\[ 0 \to H_2(K) \xrightarrow{\partial_*} H_1(U \cap V) \xrightarrow{\Phi_1} H_1(U) \oplus H_1(V) \xrightarrow{\Psi_1} H_1(K) \xrightarrow{\partial_*} H_0(U \cap V) \xrightarrow{\Phi_0} H_0(U) \oplus H_0(V) \to H_0(K) \to 0. \]
Mayer–Vietoris long exact sequence.
3.2. Second homology \(H_2(K)\):Since \(\Phi_1(n) = (2n, -2n) = (0, 0) \iff n = 0\), \(\Phi_1\) is injective (\(\ker \Phi_1 = 0\)).By exactness at \(H_1(U \cap V)\):\[ H_2(K) \cong \ker(\Phi_1) = 0. \]
injectivity of \(\Phi_1\).
3.3. Zeroth homology \(H_0(K)\):\(K\) is connected, so \(H_0(K) \cong \mathbb{Z}\).
path-connectedness of \(K\).
3.4. First homology \(H_1(K)\):The map \(\Phi_0: \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}\) given by \(1 \mapsto (1, -1)\) is injective, so \(\ker(\Phi_0) = 0\).Thus the boundary map \(\partial_*: H_1(K) \to H_0(U \cap V)\) is the zero map.By exactness:\[ H_1(K) \cong \operatorname{coker}(\Phi_1) = \frac{\mathbb{Z} \oplus \mathbb{Z}}{\operatorname{span}\{(2, -2)\}}. \]Using the basis change \(u = (1, 0)\) and \(v = (1, -1)\) for \(\mathbb{Z} \oplus \mathbb{Z}\), the submodule is generated by \(2v\).Therefore:\[ H_1(K) \cong \frac{\mathbb{Z} u \oplus \mathbb{Z} v}{2\mathbb{Z} v} \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}. \]
Smith normal form / quotient of free abelian groups.
4. Conclusion:The homology groups of the Klein bottle are:\[ H_0(K) \cong \mathbb{Z}, \qquad H_1(K) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}, \qquad H_2(K) \cong 0, \qquad H_k(K) \cong 0 \quad (k \ge 3). \]Q.E.D.
2 and 3.