Lefschetz number

An invariant of a mapping of a chain (cochain) complex or topological space into itself. Let $X$ be a chain complex of Abelian groups (respectively, a topological space), $f : X \rightarrow X$ an endomorphism of degree 0 (respectively, a continuous mapping; cf. Degree of a mapping), $H _ {i} ( X , \mathbf Q )$ the homology group of the object $X$ with coefficients in the field of rational numbers $\mathbf Q$, where

$$\sum _ { i } \mathop{\rm dim} _ {\mathbf Q } H _ {i} ( X , \mathbf Q ) < \infty ,$$

and let $t _ {i}$ be the trace of the linear transformation

$$f _ {*} : H _ {i} ( X , \mathbf Q ) \rightarrow H _ {i} ( X , \mathbf Q ) .$$

By definition, the Lefschetz number of $f$ is

$$\Lambda (f) = \sum_{i=0}^\infty (-1)^i t_i .$$

In the case of a cochain complex the definition is similar. In particular, the Lefschetz number of the identity mapping $e _ {X}$ is equal to the Euler characteristic $\chi ( X)$ of the object $X$. If $X$ is a chain (cochain) complex of free Abelian groups or a topological space, then the number $\Lambda ( f )$ is always an integer. The Lefschetz number was introduced by S. Lefschetz [1] for the solution of the problem on the number of fixed points of a continuous mapping (see Lefschetz formula).

To find the Lefschetz number of an endomorphism $f$ of a complex $X$ consisting of finite-dimensional vector spaces $X _ {i}$ over $\mathbf Q$ one can use the following formula (which is sometimes called the Hopf trace formula):

$$\Lambda ( f ) = \sum _ {i=0} ^ \infty (- 1) ^ {i} T _ {i} ,$$

where $T _ {i}$ is the trace of the linear transformation $f : X _ {i} \rightarrow X _ {i}$. In particular, if $X$ is a finite cellular space, $\phi : X \rightarrow X$ is a continuous mapping of it into itself and $\psi : X \rightarrow X$ is a cellular approximation of $\phi$, then

$$\Lambda ( \phi ) = \Lambda ( \psi ) = \sum _ {i=0}^ \infty (- 1) ^ {i} T _ {i} ,$$

where $T _ {i}$ is the trace of the transformation

$$\psi _ {\#} : C _ {i} ( X , \mathbf Q ) \rightarrow C _ {i} ( X , \mathbf Q )$$

induced by $\psi$ and $C _ {i} ( X _ {i} , \mathbf Q )$ is the group of rational $i$- dimensional chains of $X$.

Everything stated above can be generalized to the case of an arbitrary coefficient field.

References

[1]	S. Lefschetz, "Intersections and transformations of complexes and manifolds" Trans. Amer. Math. Soc. , 28 (1926) pp. 1–49
[2]	H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German)
[a1]	J. Dugundji, A. Granas, "Fixed point theory" , PWN (1982)