Contents

# Contents

## 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,

(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.

### For ordinary cohomology

The original article is

Review:

### For étale cohomology

For étale cohomology of schemes:

For algebraic stacks: