Үздіксіз бейнелеудің түрақты нүктелерін санау: Лефшец теоремасы

Counts the fixed points of a continuous mapping from a compact topological space to itself

In mathematics, the Lefschetz fixed point theorem 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 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).

First, by applying the simplicial approximation theorem, one shows that if has no fixed points, then (possibly after subdividing ) 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 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.

The Lefschetz number 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, which states that every continuous map from the dimensional closed unit disk to must have at least one fixed point. This can be seen as follows: is compact and triangulable, all its homology groups except are zero, and every continuous map induces the identity map , whose trace is one; all this together implies that is non zero for any continuous map .