nLab Lefschetz trace formula

Idea

A formula for the number of fixed points of continuous endo-maps of topological spaces.

Statement

Definition

(Lefschetz number)
Given a continuous endomorphism $f \colon X\to X$ of a topological space, its Lefschetz number is the alternating sum of the traces

$$\Lambda_k(X,f) \;\coloneqq\; \sum_i \, (-1)^i \cdot Tr \bigg( H^i(X ,\, \mathbb{Q}) \xrightarrow{ f^\ast } H^i(X ,\, \mathbb{Q}) \bigg) \,,$$

of the linear endomorphisms on the rational cohomology groups with coefficients given by pullback in cohomology .

One sometimes also speaks of the Lefschetz number of the induced endomorphism of the chain/cochain complexes, see algebraic Lefschetz formula.

Proposition

(Lefschetz fixed point theorem)
If $X$ is a compact polyhedron (a finite simplicial complex) and if the Lefschetz number (Def. ) is non-zero, then $f$ has at least one fixed point. I

The proof follows from the existence of

1. a good cycle map,

2. the Künneth formula,

3. Poincaré duality.

(see Milne, section 25).

Remark

The Lefschetz formula holds more generally in Weil cohomology theories (by definition) and hence notably in ℓ-adic étale cohomology. This fact serves to prove the Weil conjectures.

Examples

Example

(Euler characteristic)
For $f = id$ is the identity map, the Lefschetz number (Def. ) reduces to the Euler characteristic of $X$ (see this Def.):

$$\Lambda(X,\,id) \,=\, \chi(X) \,.$$

Example

(fixed point theorem for contractible polyhedra)
If $X$ is compact polyhedron which is contractible then all its cohomology groups in positive degree vanish, $H^{\geq 1}(X; k) \,=\, 0$, while $H^0(X; k) \,=\, k$. This implies that the Lefschetz number (Def. ) of every endo-map $f$ is $\Lambda(X,f) \,=\, 1$.

Therefore, for contactible compact polyhedral $X$ the Lefschetz fixed point theorem (Prop. ) says that every map $f \colon X\to X$ has a fixed point.

In the further special case that $X = D^n$ is a disk/ball, this is also the statement of Brouwer's fixed point theorem.

Example

(fixed point theorem for homeomorphisms of n-spheres)
For $n \in \mathbb{N}$, the n-sphere has ordinary cohomology, in particular, rational cohomology, concentrated in degrees 0 and $n$:

$$H^k(S^n;\, \mathbb{Q}) \;\simeq\; \left\{ \array{ \mathbb{Q} &\vert& k \in \{0,n\} \\ 0 &\vert& \text{otherwise} } \right. \,.$$

Any map $f \,\colon\, S^n \to S^n$ necessarily induces the identity on $H^0(S^n;\mathbb{Q})$ and is multiplication by the degree $d(f)$ on $H^n(S^n; \mathbb{Q})$.

Therefore the Lefschetz number of $f$ (Def. ) is

$$\Lambda(S^n, f) \;=\; 1 + (-1)^n d(f)$$

and so the Lefschetz fixed point theorem (Prop. ) in this case implies that sufficient conditions for $f$ to have a fixed point is that

• $f$ is not a homeomorphism (its degree is different from $\pm 1$);

• $f$ is a homeomorphism and

  $$d(f) \,=\, (-1)^n \,.$$

The last case means equivalently that a homeomorphism $f \,\colon\, S^n \to S^n$ is guaranteed to have a fixed point if

• $n$ is even and $f$ preserves orientation;

or

• $n$ is odd and $f$ reverses orientation.

This plays a role in the classification of free transformation group actions on n-spheres, see at group actions on spheres the section Free actions by finite groups.

For example, the antipodal reflection action of $\mathbb{Z}/2$ on $S^n$ is orientation reversing for even $n$ and orientation preserving for odd $n$ and hence always has vanishing Lefschetz number. Indeed, this action is manifestly fixed-point free. But the Lefschetz fixed point theorem implies that this is the only case, in that there can be no orientation-preserving free action of $\mathbb{Z}/2$ on an even-dimensional sphere, and no orientation-reversing free action of $\mathbb{Z}/2$ on an odd-dimensional sphere.

Related concepts

• fixed point

• Brouwer's fixed point theorem

• Atiyah-Bott fixed point formula

• Lawvere's fixed point theorem

• Lefschetz fixed point theorem

• Kleene's fixed point theorem

• Reidemeister trace

• Weil conjecture

• Grothendieck-Lefschetz trace formula

References

For ordinary cohomology

The original article is

• Solomon Lefschetz, On the fixed point formula, Ann. of Math. (2), 38 (1937) 819–822 (jstor:1968838)

Review:

• Edgar Lin, An overview and proof of the Lefschetz fixed-point theorem (pdf, pdf)

• Online Springer Enc. of Math. (eom): Lefschetz number (Rudyak), Lefschetz formula (Iskovskikh)

In equivariant cohomology:

• Minhyong Kim, A Lefschetz trace formula for equivariant cohomology, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 28 no. 6 (1995), p. 669-688 (numdam:ASENS_1995_4_28_6_669_0, MR97d:55012)

For étale cohomology

For étale cohomology of schemes:

• James Milne, section 25 of Lectures on Étale Cohomology

For algebraic stacks:

• Kai Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112, 1 (1993), 127-149 (doi:10.1007/BF01232427)