In mathematics, the Lefschetz fixed-point theorem[1] is a formula that counts the fixed points of a continuous mapping from a compact topological space
to itself by means of traces of the induced mappings on the homology groups of
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).
For a formal statement of the theorem, let be a continuous map from a compact triangulable space
by the alternating (finite) sum of the matrix traces of the linear maps induced by
A simple version of the Lefschetz fixed-point theorem states: if then
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
Note however that the converse is not true in general:
has fixed points, as is the case for the identity map on odd-dimensional spheres.
First, by applying the simplicial approximation theorem, one shows that if
has no fixed points, then (possibly after subdividing
This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of
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).
denotes the index of the fixed point
[3] From this theorem one deduces the Poincaré–Hopf theorem for vector fields, since every vector field on compact differential manifold induce flow
is continuous mapping homotopic to identity (thus have same Lefschetz number) and for small
The Lefschetz number[2] of the identity map on a finite CW complex can be easily computed by realizing that each
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
Thus we have The Lefschetz fixed-point theorem generalizes the Brouwer fixed-point theorem,[4] which states that every continuous map from the
is compact and triangulable, all its homology groups except
Lefschetz presented his fixed-point theorem in [1].
of the same dimension, the Lefschetz coincidence number of
on the cohomology groups with rational coefficients, and
Lefschetz proved that if the coincidence number is nonzero, then
be the identity map gives a simpler result, which is now known as the fixed-point theorem.
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
is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius
The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.