Lefschetz fixed-point theorem

In the Lefschetz fixed point theorem is a topological set, according to the certain continuous maps the existence of a fixed point is assured. Basis of Solomon Lefschetz theorem proved in 1926 is called the Lefschetz number at which it is a characteristic of continuous maps, which is defined using relatively abstract concepts of algebraic topology and homotopy invariant.

A worsening of the fixed-point theorem, the fixed point formula of Lefschetz, in which the Lefschetz number is expressed as a sum over fixed point indices. As a special case of Lefschetz'schen fixed point theorem yields the fixed point theorem of Brouwer and a far-reaching generalization of this set is the fixed point theorem of Atiyah and Bott in the field of Global Analysis.

Lefschetz number

The Lefschetz number can be used for every continuous self-map

Define a topological space X, all of whose Betti numbers, which are the dimensions of the vector spaces -conceived as singular homology groups are finite:

The summands of the alternating sum is the trace of the homology groups induced by f homomorphisms. Lefschetz numbers are always integers. Due to their definition, they do not change the transition to a homotopic mapping.

The Lefschetz number of the identity mapping is equal to the Euler characteristic

Lefschetz fixed point theorem

For example, in the case that the topological space has a finite triangulation K (it is then particularly compact), the Lefschetz number can already be calculated on the level of the associated finite chain complex. Specifically applies to a simplicial approximation of the mapping f fK the so-called Lefschetz - Hopf trace formula

At a fixed point free self- map f, ie, a mapping f with no points x with f ( x ) = x, can then be detected by a sufficiently refined triangulation = 0.

Conversely He has f ≠ 0 have a Lefschetz number at least a fixed point each self-image. This is the statement of the fixed point theorem of Lefschetz.

Fixed point formula of Lefschetz

Lefschetz the number of an image is dependent only on their behavior in environments of the checkpoint components. Does the map f only isolated fixed points, the Lefschetz number can be obtained by the formula

Be expressed. In this case, Fix ( f ) denotes the finite set of isolated fixed points and i ( f, x ) the fixed point index to the fixed point x.

The fixed point index can be interpreted as multiplicity of the relevant fixed point: If x is a fixed point situated inside a polyhedron X, then its fixed point index i (f, x ) is equal to the degree of image defined on a small sphere around x Figure

The Brouwer fixed point theorem as a special case

Since in the closed n-dimensional unit ball Dn for all k ≥ 1, the homology groups Hk (Dn, Q) vanish, the Lefschetz number is any self-map on Dn is 1 Each such mapping must therefore have at least one fixed point.