Show that \(S^1 \times I \simeq M\), the Mobius strip.
Solution
Goal: Prove that the cylinder \(S^1 \times [0, 1]\) is homotopy equivalent to the Möbius strip \(M = ([0, 1] \times [0, 1]) / ((0, y) \sim (1, 1-y))\).
1. Show that the cylinder \(S^1 \times [0, 1]\) deformation retracts to the circle \(S^1\).
1.1. Realize \(S^1 \times [0, 1]\) with inclusion \(\iota_1 \colon S^1 \hookrightarrow S^1 \times [0, 1]\) given by \(z \mapsto (z, 1/2)\), and retraction \(r_1 \colon S^1 \times [0, 1] \to S^1\) given by \((z, t) \mapsto z\).
1.2. Define the straight-line homotopy \(H_1 \colon (S^1 \times [0, 1]) \times [0, 1] \to S^1 \times [0, 1]\) by \(H_1((z, t), s) = (z, (1-s)t + s(1/2))\).
1.3. \(H_1\) is continuous, \(H_1((z, t), 0) = (z, t) = \operatorname{id}_{S^1 \times I}\), \(H_1((z, t), 1) = (z, 1/2) = \iota_1(r_1(z, t))\), and \(H_1((z, 1/2), s) = (z, 1/2)\) for all \(s \in [0, 1]\).
1.4. Thus \(H_1\) is a strong deformation retraction of \(S^1 \times [0, 1]\) onto the circle \(S^1 \times \{1/2\} \cong S^1\).
1.5. Consequently, \(S^1 \times [0, 1] \simeq S^1\).
1.6. The three identities in 3 are exactly the defining conditions of a strong deformation retraction, so \(H_1\) is a strong deformation retraction of \(S^1 \times [0, 1]\) onto \(S^1 \times \{1/2\} \cong S^1\).
2. Show that the Möbius strip \(M\) deformation retracts to the circle \(S^1\).
2.1. Let \(M = ([0, 1] \times [0, 1]) / ((0, y) \sim (1, 1-y))\), and let \(C = \{[(x, 1/2)] \mid x \in [0, 1]\} \subset M\) be the core circle.
2.2. The map \(\gamma \colon [0, 1] / (0 \sim 1) \to C\) given by \(x \mapsto [(x, 1/2)]\) is a homeomorphism, so \(C \cong S^1\).
2.3. Define \(H_2 \colon M \times [0, 1] \to M\) by \(H_2([(x, y)], s) = [(x, (1-s)y + s(1/2))]\).
2.4. \(H_2\) respects the quotient identification because at \(x = 1\), \(H_2([(1, 1-y)], s) = [(1, (1-s)(1-y) + s/2)] = [(0, 1 - ((1-s)(1-y) + s/2))] = [(0, (1-s)y + s/2)] = H_2([(0, y)], s)\).
2.5. \(H_2\) is a strong deformation retraction of \(M\) onto the core circle \(C \cong S^1\).
2.6. Consequently, \(M \simeq S^1\).
2.7. The map \(H_2\) is well-defined by 4 and satisfies the defining conditions of a strong deformation retraction, so it is a strong deformation retraction of \(M\) onto the core circle \(C \cong S^1\).
3. Combine homotopy equivalences.
3.1. Homotopy equivalence (\(\simeq\)) is an equivalence relation on topological spaces.
3.2. By 1, \(S^1 \times [0, 1] \simeq S^1\).
3.3. By 2, \(M \simeq S^1\).
3.4. By transitivity and symmetry, \(S^1 \times [0, 1] \simeq M\).
3.5. Since \(\simeq\) is an equivalence relation, \(S^1 \times [0, 1] \simeq S^1\) and \(M \simeq S^1\) together imply \(S^1 \times [0, 1] \simeq M\).
4. Q.E.D.
4.1. 3 establishes \(S^1 \times [0, 1] \simeq M\).