The Klein bottle \(K\) is the space obtained from a square by means of the labelling scheme \(aba^{-1}b\). Figure 74.11 indicates how \(K\) can be pictured as an immersed surface in \(\mathbb{R}^3\).
Find a presentation for the fundamental group of \(K\).
Find a double covering map \(p: T \to K\), where \(T\) is the torus. Describe the induced homomorphism of fundamental groups.
Solution
Goal: Compute the fundamental group of the Klein bottle \(K\) and construct its 2-sheeted orientable torus covering \(p: T \to K\).
1. Part (a): Presentation of \(\pi_1(K)\).Proof:
1.1. In the square \(P = [0, 1] \times [0, 1]\) with labelling scheme \(a b a^{-1} b\), the four vertices are identified to a single equivalence class \(x_0 \in K\).
1.2. The 1-skeleton \(K^1\) is a wedge of two circles \(S^1 \vee S^1\) represented by the loops \(a\) and \(b\), with \(\pi_1(K^1, x_0) \cong \langle a, b \rangle\).
1.3. The single 2-cell is attached along the boundary loop \(a b a^{-1} b\).
1.4. By the Seifert-van Kampen theorem / cellular presentation of fundamental groups,\[\pi_1(K, x_0) \cong \langle a, b \mid a b a^{-1} b = 1 \rangle \cong \langle a, b \mid a b a^{-1} = b^{-1} \rangle.\]
2. Part (b): Construction of the 2-fold covering \(p: T \to K\) and the induced map \(p_*\).Proof:
2.1. Universal covering action: The group \(G = \pi_1(K)\) acts properly discontinuously on the universal cover \(\mathbb{R}^2\) by affine isometries:\[\tau_a(x, y) = (x + 1, -y), \qquad \tau_b(x, y) = (x, y + 1).\]Note that \(\tau_a \circ \tau_b \circ \tau_a^{-1}(x, y) = (x, y - 1) = \tau_b^{-1}(x, y)\), satisfying the relation \(a b a^{-1} = b^{-1}\).
2.2. The orientation-preserving subgroup: Consider the subgroup \(H \le G\) of words containing an even exponent sum in \(a\):\[H = \langle a^2, b \rangle \triangleleft G.\]
2.3. Commutativity in \(H\):\[a^2 b a^{-2} = a (a b a^{-1}) a^{-1} = a b^{-1} a^{-1} = (a b a^{-1})^{-1} = (b^{-1})^{-1} = b.\]Thus \(a^2 b = b a^2\), and \(H \cong \mathbb{Z}^2\) is abelian of rank 2.
2.4. Identification of the quotient \(\mathbb{R}^2 / H\): The translations \(\tau_a^2(x, y) = (x + 2, y)\) and \(\tau_b(x, y) = (x, y + 1)\) generate a standard 2-dimensional lattice \(2\mathbb{Z} \times \mathbb{Z}\). The quotient space \(\mathbb{R}^2 / H\) is homeomorphic to the torus \(T = S^1 \times S^1\).
2.5. The double cover \(p: T \to K\): Since \([G : H] = 2\), the canonical projection \(p: \mathbb{R}^2 / H \to \mathbb{R}^2 / G\) is a 2-sheeted covering map from the torus \(T\) to the Klein bottle \(K\).
2.6. Induced homomorphism \(p_*\): Identifying \(\pi_1(T) \cong \langle \alpha, \beta \mid \alpha\beta = \beta\alpha \rangle\), the induced homomorphism \(p_*: \pi_1(T) \hookrightarrow \pi_1(K)\) is the inclusion of the index-2 subgroup \(H\), given by:\[p_*(\alpha) = a^2, \qquad p_*(\beta) = b.\]Q.E.D.