Identify \(\pi_1(M, [(1, \frac{1}{2})])\), and identify the class of \(\partial M\).
Solution
Goal: Let \(M = ([0, 1] \times [0, 1]) / ((0, y) \sim (1, 1-y))\) be the standard Möbius strip with basepoint \(x_0 = [(1, 1/2)] = [(0, 1/2)]\).
Compute the fundamental group \(\pi_1(M, x_0)\).
Identify the homotopy class \([\partial M] \in \pi_1(M, x_0)\) represented by its boundary circle.
1. Compute \(\pi_1(M, x_0)\).
1.1. Define the center circle / core curve \(\gamma_0 \colon [0, 1] \to M\) by \(\gamma_0(t) = [(t, 1/2)]\).
1.2. \(\gamma_0(0) = [(0, 1/2)] = [(1, 1/2)] = \gamma_0(1) = x_0\), so \(\gamma_0\) is a continuous loop based at \(x_0\).
1.3. Let \(C = \gamma_0([0, 1]) \subset M\) be the core circle, homeomorphic to \(S^1\).
1.4. Define \(H \colon M \times [0, 1] \to M\) by \(H([(x, y)], s) = [(x, (1-s)y + s(1/2))]\).
1.5. \(H\) is well-defined: \(H([(1, 1-y)], s) = [(1, (1-s)(1-y) + s/2)] = [(1, 1 - ((1-s)y + s/2))] = H([(0, y)], s)\).
1.6. \(H\) is a strong deformation retraction of \(M\) onto the core circle \(C\).
1.7. Thus \(\pi_1(M, x_0) \cong \pi_1(C, x_0) \cong \mathbb{Z}\), generated by the homotopy class \([\gamma_0]\).
1.8. The map \(H\) is a strong deformation retraction of \(M\) onto the core circle \(C \cong S^1\), and a deformation retraction induces an isomorphism on fundamental groups, so \(\pi_1(M, x_0) \cong \pi_1(S^1) \cong \mathbb{Z}\).
2. Identify the homotopy class of the boundary \(\partial M\).
2.1. The boundary of \(M\) comes from the horizontal edges \(y = 0\) and \(y = 1\) in \([0, 1] \times [0, 1]\).
2.2. Under the gluing \((0, y) \sim (1, 1-y)\), \((0, 0) \sim (1, 1)\) and \((0, 1) \sim (1, 0)\).
2.3. A continuous parameterization of the single boundary loop \(\partial M\) starting and ending at \(p_0 = [(0, 0)] = [(1, 1)]\) is given by: \[\beta(t) = \begin{cases} [(2t, 0)] & \text{for } t \in [0, 1/2], \\ [(2t-1, 1)] & \text{for } t \in [1/2, 1]. \end{cases}\]
2.4. To compare with basepoint \(x_0 = [(0, 1/2)]\), connect \(x_0\) to \(p_0\) by the straight segment \(\eta(s) = [(0, (1-s)/2)]\) for \(s \in [0, 1]\).
2.5. The based boundary loop is \(\alpha = \eta \cdot \beta \cdot \eta^{-1}\).
2.6. Applying the deformation retraction \(H\) from 1 to \(\beta\):
On \(t \in [0, 1/2]\), \(H([(2t, 0)], 1) = [(2t, 1/2)] = \gamma_0(2t)\), which traverses the core loop once.
On \(t \in [1/2, 1]\), \(H([(2t-1, 1)], 1) = [(2t-1, 1/2)] = \gamma_0(2t-1)\), which traverses the core loop a second time in the same direction. <2>7. Thus \(H(\alpha, 1) \simeq \gamma_0 \cdot \gamma_0 = \gamma_0^2\). <2>8. Therefore, the class of the boundary loop is \([\partial M] = [\gamma_0]^2 = 2 \in \mathbb{Z} \cong \pi_1(M, x_0)\) (i.e. twice the generator).
2.9. Applying the deformation retraction \(H\) to the based boundary loop \(\alpha\) sends it to \(\gamma_0 \cdot \gamma_0 = \gamma_0^2\), and \(H\) is a homotopy equivalence, so \([\partial M] = [\gamma_0]^2 = 2\) in \(\pi_1(M, x_0) \cong \mathbb{Z}\).
3. Conclusion.
3.1. \(\pi_1(M, x_0) \cong \mathbb{Z} = \langle a \rangle\), and \([\partial M] = a^2 = 2 \in \mathbb{Z}\).
3.2. 1 identifies \(\pi_1(M, x_0)\) with \(\mathbb{Z}\) generated by \(a = [\gamma_0]\), and 2 shows \([\partial M] = a^2\).