Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.

The counting is subject to an imputed multiplicity at a fixed point called the fixed point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).

Formal statement

For a formal statement of the theorem, let

$f\colon X\rightarrow X\,$ be a continuous map from a compact triangulable space $X$ to itself. Define the Lefschetz number $\Lambda _{f}$ of $f$ by

$\Lambda _{f}:=\sum _{k\geq 0}(-1)^{k}\mathrm {tr} (f_{*}|H_{k}(X,\mathbb {Q} )),$ the alternating (finite) sum of the matrix traces of the linear maps induced by $f$ on $H_{k}(X,\mathbb {Q} )$ , the singular homology groups of $X$ with rational coefficients.

A simple version of the Lefschetz fixed-point theorem states: if

$\Lambda _{f}\neq 0\,$ then $f$ has at least one fixed point, i.e., there exists at least one x in $X$ such that $f(x)=x$ . In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to $f$ has a fixed point as well.

Note however that the converse is not true in general: $\Lambda _{f}$ may be zero even if f has fixed points.

Sketch of a proof

First, by applying the simplicial approximation theorem, one shows that if $f$ has no fixed points, then (possibly after subdividing X) $f$ is homotopic to a fixed-point-free simplicial map (i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of X must be all be zero. Then one notes that, in general, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic). In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.

Lefschetz–Hopf theorem

A stronger form of the theorem, also known as the Lefschetz–Hopf theorem, states that, if $f$ has only finitely many fixed points, then

$\sum _{x\in \mathrm {Fix} (f)}i(f,x)=\Lambda _{f},$ where $\mathrm {Fix} (f)$ is the set of fixed points of $f$ , and $i(f,x)$ denotes the index of the fixed point x. From this theorem one deduces the Poincaré–Hopf theorem for vector fields.

Relation to the Euler characteristic

The Lefschetz number of the identity map on a finite CW complex can be easily computed by realizing that each $f_{\ast }$ can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic $\chi (X)$ . Thus we have

$\Lambda _{\mathrm {id} }=\chi (X).\$ Relation to the Brouwer fixed point theorem

The Lefschetz fixed point theorem generalizes the Brouwer fixed point theorem, which states that every continuous map from the n-dimensional closed unit disk $D^{n}$ to $D^{n}$ must have at least one fixed point.

This can be seen as follows: $D^{n}$ is compact and triangulable, all its homology groups except $H_{0}$ are zero, and every continuous map $f\colon D^{n}\to D^{n}$ induces the identity map $f_{*}\colon H_{0}(D^{n},\mathbb {Q} )\to H_{0}(D^{n},\mathbb {Q} )$ , whose trace is one; all this together implies that $\Lambda _{f}$ is non-zero for any continuous map $f\colon D^{n}\to D^{n}$ .

Historical context

Lefschetz presented his fixed point theorem in (Lefschetz 1926). Lefschetz's focus was not on fixed points of maps, but rather on what are now called coincidence points of maps.

Given two maps f and g from an orientable manifold X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as

$\Lambda _{f,g}=\sum (-1)^{k}\mathrm {tr} (D_{X}\circ g^{*}\circ D_{Y}^{-1}\circ f_{*}),$ where $f_{*}$ is as above, $g_{*}$ is the homomorphism induced by $g$ on the cohomology groups with rational coefficients, and $D_{X}$ and $D_{X}$ are the Poincaré duality isomorphisms for X and Y, respectively.

Lefschetz proved that if the coincidence number is nonzero, then f and g have a coincidence point. He noted in his paper that letting $X=Y$ and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.

Frobenius

Let $X$ be a variety defined over the finite field $k$ with $q$ elements and let ${\bar {X}}$ be the lift of $X$ to the algebraic closure of $k$ . The Frobenius endomorphism of ${\bar {X}}$ (often the geometric Frobenius, or just the Frobenius), denoted by $F_{q}$ , maps a point with coordinates $x_{1},\ldots ,x_{n}$ to the point with coordinates $x_{1}^{q},\ldots ,x_{n}^{q}$ . Thus the fixed points of $F_{q}$ are exactly the points of $X$ with coordinates in $k$ ; the set of such points is denoted by $X(k)$ . The Lefschetz trace formula holds in this context, and reads:

$\#X(k)=\sum _{i}(-1)^{i}\mathrm {tr} F_{q}|H_{c}^{i}({\bar {X}},\mathbb {Q} _{\ell }).$ This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of ${\bar {X}}$ with values in the field of $\ell$ -adic numbers, where $\ell$ is a prime coprime to $q$ .

If $X$ is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius $\Phi _{q}$ , which acts as the inverse of $F_{q}$ on cohomology:

$\#X(k)=q^{\dim X}\sum _{i}(-1)^{i}\mathrm {tr} \Phi _{q}^{-1}|H^{i}({\bar {X}},\mathbb {Q} _{\ell }).$ This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.