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A torus translation has zero Lefschetz number and no fixed points

Example

Assume AC (The Axiom of Choice). Let $T^{2} = R^{2} / Z^{2}$ be the two-torus and let $T_{a} : T^{2} \to T^{2}$ , $T_{a} (x) = x + a$ , be the translation by $a \in R^{2}$ . If $a \neq 0$ in $R^{2} / Z^{2}$ then $T_{a}$ has no fixed points; and $L (T_{a}) = 0 = χ (T^{2})$ (Algebraic Lefschetz number via rational homology traces). This realises the sharpness of the nonzero hypothesis in the Lefschetz fixed point theorem: the conclusion of Lefschetz fixed point theorem can fail when $L = 0$ . This example has $L (T_{a}) = χ (T^{2}) = 0$ and no fixed points; the separate counterexample with canceling fixed points refutes the converse.

Verification

Given: The torus $T^{2} = R^{2} / Z^{2}$ and a translation $T_{a}$ by a nonzero class $a$ .

[F1] Present the torus as the square with opposite edges identified. Its vertices form one point, its open horizontal and vertical edges form two $1$ -cells, and its open interior is one $2$ -cell. Traversing the boundary gives the attaching word $a b a^{- 1} b^{- 1}$ . Thus this CW structure has one $0$ -cell, two $1$ -cells and one $2$ -cell attached along the commutator $a b a^{- 1} b^{- 1}$ ; with this structure the alternating cell count is $1 - 2 + 1 = 0$ (CW complex with closure finiteness and weak topology, Cell attachment by a characteristic map, Euler characteristic of a finite CW complex).

[F2] On a compact manifold the Euler characteristic is the alternating sum of the rational Betti numbers, and it agrees with the cell count for a finite CW model (Euler characteristic of a compact manifold); clause (ii) of Finiteness and additivity of the Euler characteristic computes it from a finite relative cell decomposition, and the translation and the identity are homotopic through translations.

[L1] Lefschetz numbers are homotopy invariant and $L ({i d}_{M}) = χ (M)$ (The Lefschetz number is a homotopy invariant, The Lefschetz number of the identity is the Euler characteristic, The Axiom of Choice).

1.1given

No fixed points for $a \neq 0$ . A point $x \in T^{2}$ is fixed by $T_{a}$ exactly when $a = x - x = 0$ in $R^{2} / Z^{2}$ ; hence for nonzero $a$ the translation is fixed-point-free.

2.1step 1.1F1F2L1∎

The Lefschetz number vanishes. The family $t \mapsto T_{t a}$ , $t \in [0, 1]$ , is a homotopy from the identity to $T_{a}$ , so $L (T_{a}) = L (i d) = χ (T^{2})$ by [L1]. The standard CW structure of [F1] has cell count $1 - 2 + 1 = 0$ , and by [F2] this is $χ (T^{2})$ ; hence $L (T_{a}) = 0$ . Thus a vanishing Lefschetz number coincides here with the complete absence of fixed points, which is exactly why the example does not contradict the theorem but exhibits its sharpness.

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Sources

Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall 1974; complete 236-page PDF) (standard reference, not scraped)
Peter Wong, Lectures on Fixed Point Theory, Mini-Course XV Encontro Brasileiro de Topologia, Rio Claro 2006 (complete notes) (standard reference, not scraped)
Allen Hatcher, Algebraic Topology (complete book PDF) (standard reference, not scraped)