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)