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 formula_1 to itself by means of traces of the induced mappings on the homology groups of formula_1. 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
be a continuous map from a compact triangulable space formula_1 to itself. Define the Lefschetz number formula_5 of formula_6 by
the alternating (finite) sum of the matrix traces of the linear maps induced by formula_6 on formula_9, the singular homology groups of formula_1 with rational coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
then formula_6 has at least one fixed point, i.e., there exists at least one formula_13 in formula_1 such that formula_15. 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 formula_6 has a fixed point as well.
Note however that the converse is not true in general: formula_5 may be zero even if formula_6 has fixed points.
Sketch of a proof.
First, by applying the simplicial approximation theorem, one shows that if formula_6 has no fixed points, then (possibly after subdividing formula_1) formula_6 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 formula_1 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 formula_6 has only finitely many fixed points, then
where formula_25 is the set of fixed points of formula_6, and formula_27 denotes the index of the fixed point formula_13. 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 formula_29 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 formula_30. Thus we have
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 formula_32-dimensional closed unit disk formula_33 to formula_33 must have at least one fixed point.
This can be seen as follows: formula_33 is compact and triangulable, all its homology groups except formula_36 are zero, and every continuous map formula_37 induces the identity map formula_38, whose trace is one; all this together implies that formula_5 is non-zero for any continuous map formula_37.
Historical context.
Lefschetz presented his fixed point theorem in . Lefschetz's focus was not on fixed points of maps, but rather on what are now called coincidence points of maps.
Given two maps formula_6 and formula_42 from an orientable manifold formula_1 to an orientable manifold formula_44 of the same dimension, the "Lefschetz coincidence number" of formula_6 and formula_42 is defined as
where formula_48 is as above, formula_49 is the homomorphism induced by formula_42 on the cohomology groups with rational coefficients, and formula_51 and formula_51 are the Poincaré duality isomorphisms for formula_1 and formula_44, respectively.
Lefschetz proved that if the coincidence number is nonzero, then formula_6 and formula_42 have a coincidence point. He noted in his paper that letting formula_57 and letting formula_42 be the identity map gives a simpler result, which we now know as the fixed point theorem.
Frobenius.
Let formula_1 be a variety defined over the finite field formula_60 with formula_61 elements and let formula_62 be the base change of formula_1 to the algebraic closure of formula_60. The Frobenius endomorphism of formula_62 (often the "geometric Frobenius", or just "the Frobenius"), denoted by formula_66, maps a point with coordinates formula_67 to the point with coordinates formula_68. Thus the fixed points of formula_66 are exactly the points of formula_1 with coordinates in formula_60; the set of such points is denoted by formula_72. The Lefschetz trace formula holds in this context, and reads:
This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of formula_62 with values in the field of formula_75-adic numbers, where formula_75 is a prime coprime to formula_61.
If formula_1 is smooth and equidimensional, this formula can be rewritten in terms of the "arithmetic Frobenius" formula_79, which acts as the inverse of formula_66 on cohomology:
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.