Use the Lefschetz fixed point theorem to show that a map \(f: S^n \to S^n\) has a fixed point unless its degree is equal to the degree of the antipodal map \(x \mapsto -x\).
Solution
Goal: Prove via the Lefschetz Fixed Point Theorem that any continuous map \(f: S^n \to S^n\) has a fixed point whenever \(\deg(f) \neq \deg(a)\), where \(a(x) = -x\) is the antipodal map.
1. Degree and Lefschetz number of the antipodal map:Proof:
1.1. The antipodal map \(a: S^n \to S^n\) is given by \(a(x_0, x_1, \dots, x_n) = (-x_0, -x_1, \dots, -x_n)\) on \(S^n \subset \mathbb{R}^{n+1}\).
1.2. The map \(a\) is the composition of \(n + 1\) coordinate reflections \(r_i(x_0, \dots, x_i, \dots, x_n) = (x_0, \dots, -x_i, \dots, x_n)\).
1.3. Each reflection has degree \(-1\), so by functoriality of topological degree:\[\deg(a) = \deg(r_0 \circ r_1 \circ \cdots \circ r_n) = (-1)^{n+1}.\]
2. Lefschetz number of an arbitrary continuous map \(f: S^n \to S^n\):Proof:
2.1. The rational homology groups of the \(n\)-sphere \(S^n\) are:\[H_k(S^n; \mathbb{Q}) \cong \begin{cases} \mathbb{Q} & k = 0, n, \\ 0 & 0 < k < n \text{ and } k > n. \end{cases}\]
2.2. For \(k = 0\), \(S^n\) is non-empty and connected, so \(f_*: H_0(S^n; \mathbb{Q}) \to H_0(S^n; \mathbb{Q})\) is the identity, with \(\operatorname{tr}(f_*|_{H_0}) = 1\).
2.3. For \(k = n\), by definition of topological degree, \(f_*: H_n(S^n; \mathbb{Q}) \to H_n(S^n; \mathbb{Q})\) is multiplication by \(\deg(f) = d\), so \(\operatorname{tr}(f_*|_{H_n}) = d\).
2.4. If \(n = 0\), \(S^0 = \{-1, 1\}\) has \(H_0(S^0; \mathbb{Q}) \cong \mathbb{Q}^2\), where a degree \(d\) map satisfies \(\Lambda(f) = 1 + d\).
2.5. For \(n \ge 1\), the Lefschetz number \(\Lambda(f)\) is\[\Lambda(f) = \sum_{k=0}^n (-1)^k \operatorname{tr}(f_*|_{H_k(S^n; \mathbb{Q})}) = (-1)^0 (1) + (-1)^n (d) = 1 + (-1)^n d.\]
3. Fixed point existence via the Lefschetz Fixed Point Theorem:Proof:
3.1. The sphere \(S^n\) is a compact, triangulable CW complex (a compact polyhedron).
3.2. By the Lefschetz Fixed Point Theorem, if \(\Lambda(f) \neq 0\), then \(f\) must have at least one fixed point (there exists \(x \in S^n\) such that \(f(x) = x\)).
3.3. Solve for when the Lefschetz number vanishes:\[\Lambda(f) = 0 \iff 1 + (-1)^n d = 0 \iff (-1)^n d = -1 \iff d = -(-1)^{-n} = (-1)^{n+1}.\]
3.4. By 1, \((-1)^{n+1} = \deg(a)\).
3.5. Therefore, \(\Lambda(f) = 0\) if and only if \(\deg(f) = \deg(a) = (-1)^{n+1}\).
3.6. If \(\deg(f) \neq \deg(a)\), then \(\Lambda(f) \neq 0\), so \(f\) must have a fixed point.
4. Conclusion:Proof:Every continuous map \(f: S^n \to S^n\) has a fixed point unless \(\deg(f) = (-1)^{n+1} = \deg(\text{antipodal})\).