In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed pointss of a continuous mapping from a compact topological space to itself by means of tracess of the induced mappings on the homology groups. 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 to itself. Define the Lefschetz number by

the alternating (finite) sum of the matrix traces of the linear maps induced, the singular homology with rational coefficients.

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

. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic has a fixed point as well.

has fixed points, as is the case for the identity map on odd-dimensional spheres.

The same conclusion could be obtained for any compact ANR, in particular any compact topological manifold. The basic ingredient behind this extension is that compact ANRs are homotopy equivalent to finite simplicial complexes.

Sketch of a proof

First, by applying the simplicial approximation theorem 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 must 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 has only finitely many fixed points, then

denotes the index. From this theorem one may deduce the Poincaré–Hopf theorem for vector fields as follows. Any vector field on a compact manifold induces a flow the fixed points of the flow and the zeroes of the vector field have the same indices.

Relation to the Euler characteristic

The Lefschetz number of the identity map on a finite CW complex 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

Relation to the Brouwer fixed-point theorem

The Lefschetz fixed-point theorem generalizes the Brouwer fixed-point theorem,-dimensional closed unit disk must have at least one fixed point.

.

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.

from an orientable manifold of the same dimension, the Lefschetz coincidence number is defined as

on the cohomology are the Poincaré duality, respectively.

be the identity map gives a simpler result, which is now known as the fixed-point theorem.

Frobenius

be a variety defined over the finite field to the algebraic closure. The Frobenius endomorphism (often the geometric Frobenius, or just the Frobenius. The Lefschetz trace formula holds in this context, and reads:

This formula involves the trace of the Frobenius on the étale cohomology with values in the field of \ell-adic numbers.

is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius 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.

See also

• Fixed-point theorems

• Lefschetz zeta function

• Holomorphic Lefschetz fixed-point formula

References