Show that \(S^3 - \{p_0, p_1\} \simeq S^2\)
Solution
Goal: Prove that the complement of two distinct points in the 3-sphere, \(S^3 \setminus \{p_0, p_1\}\), is homotopy equivalent to \(S^2\).
1. \(S^3 \setminus \{p_0\}\) is homeomorphic to \(\mathbb{R}^3\).
1.1. Standard stereographic projection from the pole \(p_0\) gives a homeomorphism \(\phi \colon S^3 \setminus \{p_0\} \to \mathbb{R}^3\).
1.2. Stereographic projection is a bijection with an explicit formula, and both it and its inverse are continuous, so it is a homeomorphism.
2. Under the homeomorphism \(\phi\), \(S^3 \setminus \{p_0, p_1\}\) is homeomorphic to \(\mathbb{R}^3 \setminus \{\phi(p_1)\}\).
2.1. A homeomorphism \(\phi \colon X \to Y\) restricts to a homeomorphism \(X \setminus \{x\} \to Y \setminus \{\phi(x)\}\) for any point \(x \in X\).Here \(X = S^3 \setminus \{p_0\}\) and \(x = p_1 \in X\) since \(p_1 \neq p_0\).
3. \(\mathbb{R}^3 \setminus \{\phi(p_1)\}\) is homeomorphic to \(\mathbb{R}^3 \setminus \{0\}\).
3.1. Translation by \(-\phi(p_1)\), given by \(v \mapsto v - \phi(p_1)\), is a homeomorphism of \(\mathbb{R}^3\) that maps \(\phi(p_1)\) to the origin \(0\).
3.2. Restricting this homeomorphism to \(\mathbb{R}^3 \setminus \{\phi(p_1)\}\) gives a homeomorphism onto \(\mathbb{R}^3 \setminus \{0\}\).
4. \(\mathbb{R}^3 \setminus \{0\}\) deformation retracts to the unit sphere \(S^2\).
4.1. Let \(r \colon \mathbb{R}^3 \setminus \{0\} \to S^2\) be defined by \(r(x) = \frac{x}{\|x\|}\) and let \(\iota \colon S^2 \hookrightarrow \mathbb{R}^3 \setminus \{0\}\) be the inclusion.
4.2. The map \(r\) is continuous and restricts to the identity on \(S^2\) because for any \(x \in S^2\), \(\|x\| = 1 \implies r(x) = x\).
4.3. Define \(H \colon (\mathbb{R}^3 \setminus \{0\}) \times [0, 1] \to \mathbb{R}^3 \setminus \{0\}\) by \(H(x, t) = (1-t)x + t \frac{x}{\|x\|}\).
4.4. For all \(x \in \mathbb{R}^3 \setminus \{0\}\) and \(t \in [0, 1]\), \((1-t) + \frac{t}{\|x\|} > 0\), so \(H(x, t) \neq 0\).Thus \(H\) is a well-defined continuous map into \(\mathbb{R}^3 \setminus \{0\}\).
4.5. \(H(x, 0) = x = \operatorname{id}_{\mathbb{R}^3 \setminus \{0\}}(x)\), \(H(x, 1) = \iota(r(x))\), and for every \(s \in S^2\) and \(t \in [0, 1]\), \(H(s, t) = s\).
4.6. The three identities in 5 are exactly the defining conditions of a strong deformation retraction: \(H(\cdot, 0) = \operatorname{id}\), \(H(\cdot, 1) = \iota \circ r\), and \(H\) fixes \(S^2\) pointwise throughout.
5. Homotopy equivalence is an equivalence relation preserved under homeomorphisms.
5.1. Every homeomorphism is a homotopy equivalence, and a deformation retraction is a homotopy equivalence.
5.2. Composing homotopy equivalences yields a homotopy equivalence: \[S^3 \setminus \{p_0, p_1\} \cong \mathbb{R}^3 \setminus \{\phi(p_1)\} \cong \mathbb{R}^3 \setminus \{0\} \simeq S^2.\]
6. Q.E.D.
6.1. Combining 1 through 5 establishes \(S^3 \setminus \{p_0, p_1\} \simeq S^2\).