Questions: The Lefschetz Fixed Point Theorem

Question 1

A map f: S^2 → S^2 has degree 3. What is its Lefschetz number, and does it have a fixed point?

• A. L(f) = 4, so f has a fixed point
• B. L(f) = 2, so f has a fixed point
• C. L(f) = 0, so the theorem gives no information
• D. L(f) = 1 - 0 + 3 = 4, so f has a fixed point

Answer: (D)

H_0(S^2) = Z (f_* is identity, trace = 1), H_1(S^2) = 0 (trace = 0), H_2(S^2) = Z (f_* is multiplication by 3, trace = 3). So L(f) = 1 - 0 + 3 = 4 ≠ 0. By the Lefschetz theorem, f has a fixed point. In fact, L(f) = 1 + (-1)^n · deg(f) = 1 + deg(f) for maps of S^n (since f_* on H_0 always has trace 1 and on H_n has trace deg(f)). For S^2: L(f) = 1 + deg(f).

Question 2

The Lefschetz number of the identity map id: X → X equals the Euler characteristic χ(X).

• T. True
• F. False

Answer: True

For the identity map, the induced map id_* on each H_n(X; Q) is the identity, whose trace equals the dimension dim_Q(H_n(X; Q)) = b_n (the n-th Betti number). So L(id) = Σ(-1)^n b_n = χ(X). This connects the Lefschetz fixed point theorem to the Euler characteristic: if χ(X) ≠ 0, then the identity map has a fixed point — which is trivially true (every point is fixed). The nontrivial content is for maps OTHER than the identity.

Question 3

The antipodal map a: S^{2k} → S^{2k} on an even-dimensional sphere has L(a) = 1 + (-1)^{2k}(-1)^{2k+1} = 1 - 1 = 0. Does this mean the antipodal map has a fixed point?

• A. Yes, L(a) = 0 forces a fixed point
• B. No, L(a) = 0 means the theorem gives no information — and indeed the antipodal map has NO fixed points
• C. Yes, because the Euler characteristic of S^{2k} is 2
• D. No, because deg(a) = -1

Answer: (B)

L(a) = 0 means the Lefschetz theorem is inconclusive — it does NOT assert that fixed points exist or don't exist. In this case, the antipodal map a(x) = -x on S^{2k} indeed has no fixed points (a(x) = x would require x = -x, only possible for x = 0 which is not on the sphere). So the theorem correctly gives no information. The converse of the Lefschetz theorem is false: L(f) = 0 does not imply f is fixed-point-free.

Question 4

A continuous map f: T^2 → T^2 on the torus induces f_*: H_1(T^2; Q) → H_1(T^2; Q), which is a 2×2 matrix A with integer entries. Express L(f) in terms of A.

Model answer: L(f) = tr(f_*|H_0) - tr(f_*|H_1) + tr(f_*|H_2). On H_0 ≅ Q, f_* is the identity (trace 1). On H_1 ≅ Q^2, f_* is the matrix A (trace = tr(A)). On H_2 ≅ Q, f_* is multiplication by det(A) (the map on the top homology of an oriented surface is multiplication by the determinant of the map on H_1). So L(f) = 1 - tr(A) + det(A).

For the torus, the formula L(f) = 1 - tr(A) + det(A) where A is the 2×2 matrix of f_* on H_1 is explicit and computable. For example, the identity has A = I, giving L = 1 - 2 + 1 = 0 — consistent with the fact that translations on the torus are fixed-point-free and homotopic to the identity (and χ(T^2) = 0). A map with A = [[2,0],[0,2]] has L = 1 - 4 + 4 = 1 ≠ 0, so it must have a fixed point.