Fixed point theorem by Lefschetz

When fixed-point theorem of Lefschetz is a topological theorem, which in certain under steady pictures the existence of a fixed point is secured. Basis of the of Solomon Lefschetz proven record in 1926 is the so-called Lefschetz number , it is when a parameter is continuous functions, the relatively abstract concepts with the help of algebraic topology is defined and a homotopy - invariant is.

A tightening of the fixed point theorem is Lefschetz's fixed point formula , in which the Lefschetz number is expressed as the sum of fixed point indices. Brouwer's Fixed Point Theorem results as a special case of Lefschetz's Fixed Point Theorem and a far-reaching generalization of this theorem is the Fixed Point Theorem of Atiyah and Bott from the field of global analysis .

Lefschetz number

The Lefschetz number can be used for any continuous self-mapping

${\displaystyle f\colon X\rightarrow X}$

define on a topological space , whose all Betti numbers , i.e. the dimensions of the singular homology groups understood as vector spaces , are finite: ${\displaystyle X}$

${\displaystyle \Lambda _{f}:=\sum _{k\geq 0}(-1)^{k}\mathrm {Tr} (f_{k}|H_{k}(X,\mathbb {Q} )),}$

The summands of the alternating sum are the traces of the homomorphisms induced on the homology groups . Lefschetz numbers are basically whole numbers. Due to their definition, they do not change on transition to a homotopic mapping. ${\displaystyle f}$ ${\displaystyle f_{k}}$

The Lefschetz number for identical mapping is equal to the Euler characteristic

${\displaystyle \chi (X)=\,\Lambda _{\mathrm {id} }.}$

Fixed point theorem by Lefschetz

For example, in the case that the topological space has a finite triangulation (it is then especially compact ), the Lefschetz number can already be calculated at the level of the assigned finite chain complex . Specifically, the so-called Lefschetz-Hopfsche trace formula applies to a simplistic approximation of the figure ${\displaystyle K}$ ${\displaystyle C_{*}(K,\mathbb {Q} )}$ ${\displaystyle f^{K}}$ ${\displaystyle f}$

${\displaystyle \Lambda _{f}=\sum _{k\geq 0}(-1)^{k}\mathrm {Tr} (f_{k}^{K}|C_{k}(K,\mathbb {Q} )).}$

In the case of a self- mapping without fixed points , that is to say a mapping without points with , it can then be detected by means of a sufficiently refined triangulation . ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f(x)=x}$ ${\displaystyle \Lambda _{f}=0}$

Conversely, every self-image with a Lefschetz number must have at least one fixed point. This is the statement of Lefschetz's fixed point theorem. ${\displaystyle f}$ ${\displaystyle \Lambda _{f}\neq 0}$

Fixed point formula from Lefschetz

The Lefschetz number of an image depends only on its behavior in the vicinity of the fixed point components. If the figure only has isolated fixed points, the Lefschetz number can be given by the formula ${\displaystyle f}$

${\displaystyle \Lambda _{f}=\sum _{x\in \operatorname {Fix} (f)}i(f,x)}$

be expressed. It denotes the finite set of isolated fixed points and the fixed point index to the fixed point . ${\displaystyle \operatorname {Fix} (f)}$ ${\displaystyle i(f,x)}$ ${\displaystyle x}$

The fixed point index can be understood as the multiplicity of the relevant fixed point: If a fixed point located inside a polyhedron , then its fixed point index is equal to the degree of mapping of the image defined on a small sphere ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle i(f,x)}$ ${\displaystyle x}$

${\displaystyle g(y)={\frac {y-f(y)}{||y-f(y)||}}.}$

Brouwer's fixed point theorem as a special case

Since in the closed -dimensional unit sphere the homology groups disappear for all , the Lefschetz number of each self-mapping is equal to 1. Each such mapping must have at least one fixed point. ${\displaystyle n}$ ${\displaystyle D^{n}}$ ${\displaystyle k\geq 1}$ ${\displaystyle H_{k}(D^{n},\mathbf {Q} )}$ ${\displaystyle D^{n}}$

Individual evidence

1. ↑ S. Lefschetz: Intersections and transformations of complexes and manifolds , Transactions American Mathematical Society 1926, Vol. 28, pp. 1-49 ( Online ; PDF; 4.3 MB)
2. ↑ Heinz Hopf : A new proof of the Lefschetz formula on invariant points , Proceedings of the National Academy of Sciences of the USA, Vol. 14 (1928), pp. 149–153 ( Online ; PDF; 421 kB)

Web links

• Algebraic topology and fixed points . Introductory overview article by Jörg Bewersdorff . (PDF file; 179 kB)

literature

• Robert F. Brown: Fixed Point Theory. In: IM James : History of Topology. Elsevier, Amsterdam et al. 1999, ISBN 0-444-82375-1 , pp. 271-299.