Lefschetz formula

A formula that expresses the number of fixed points of an endomorphism of a topological space in terms of the traces of the corresponding endomorphisms in the cohomology groups.

This formula was first established by S. Lefschetz for finite-dimensional orientable topological manifolds [1] and for finite cell complexes (see [2], [3]). These papers of Lefschetz were preceded by a paper of L.E.J. Brouwer (1911) on the fixed point of a continuous mapping of an $ n $- dimensional sphere into itself. A new version of the proof of the Lefschetz formula for finite cell complexes was given by H. Hopf (see [9]).

Let $ X $ be a connected orientable $ n $- dimensional compact topological manifold or an $ n $- dimensional finite cell complex, let $ f : X \rightarrow X $ be a continuous mapping and let $ \Lambda ( f , X ) $ be the Lefschetz number of $ f $. Assume that all fixed points of the mapping $ f : X \rightarrow X $ are isolated. For each fixed point $ x \in X $, let $ i ( x) $ be its Kronecker index (the local degree (cf. Degree of a mapping) of $ f $ in a neighbourhood of $ x $). Then the Lefschetz formula for $ X $ and $ f $ has the form

$$ \tag{1 } \sum _ {f ( x) = x } i ( x) = \Lambda ( f , X ) . $$

There is, [8], a generalization of the Lefschetz formula to the case of arbitrary continuous mappings of compact Euclidean neighbourhood retracts.

Let $ X $ be a differentiable compact orientable manifold and let $ f : X \rightarrow X $ be a differentiable mapping. A fixed point $ x \in X $ for $ f $ is said to be non-singular if it is isolated and if $ \mathop{\rm det} ( df _ {x} - E ) \neq 0 $, where $ df _ {x} : T _ {x} ( X) \rightarrow T _ {x} ( X) $ is the differential of $ f $ at $ x $ and $ E $ is the identity transformation. For a non-singular point $ x \in X $ its index $ i ( x) $ coincides with the number $ \mathop{\rm sgn} \mathop{\rm det} ( df _ {x} - E ) $. In this case the Lefschetz formula (1) shows that the Lefschetz number $ \Lambda ( f , X ) $ is equal to the difference between the number of fixed points with index $ + 1 $ and the number of fixed points with index $ - 1 $; in particular, it does not exceed the total number of fixed points. In this case the left-hand side of (1) can be determined in the same way as the intersection index $ \Gamma _ {f} \Delta $ on $ X \times X $, where $ \Gamma _ {f} $ is the graph of $ f $ and $ \Delta \subset X \times X $ is the diagonal (cf. Intersection index (in algebraic geometry)).

A consequence of the Lefschetz formula is the Hopf formula, which asserts that the Euler characteristic $ \chi ( X) $ is equal to the sum of the indices of the zeros of a global $ C ^ \infty $- vector field $ v $ on $ X $( it is assumed that all zeros of $ v $ are isolated) (see [5]).

There is a version of the Lefschetz formula for compact complex manifolds and the Dolbeault cohomology (see [5]). Let $ X $ be a compact complex manifold of dimension $ m $ and let $ f : X \rightarrow X $ a be holomorphic mapping with non-singular fixed points. Let $ H ^ {p,q} ( X) $ be the Dolbeault cohomology of $ X $ of type $ ( p , q ) $( cf. Differential form) and let $ f ^ { * } : H ^ {p,q} ( X) \rightarrow H ^ {p,q} ( X) $ be the endomorphism induced by $ f $. The number

$$ \Lambda ( f , {\mathcal O} _ {X} ) = \sum _ { q= } 0 ^ { m } (- 1) ^ {q} \mathop{\rm Tr} ( f ^ { * } ; H ^ {0,q} ( X) ) $$

is called the holomorphic Lefschetz number. One then has the following holomorphic Lefschetz formula:

$$ \Lambda ( f , {\mathcal O} _ {X} ) = \sum _ {f ( x) = x } \frac{1}{ \mathop{\rm det} ( E - df _ {x} ) } , $$

where $ df _ {x} $ is the holomorphic differential of $ f $ at $ x $.

In abstract algebraic geometry the Lefschetz formula has served as a starting point in the search for Weil cohomology in connection with Weil's conjectures about zeta-functions of algebraic varieties defined over finite fields (cf. Zeta-function). An analogue of the Lefschetz formula in abstract algebraic geometry has been established for $ l $- adic cohomology with compact support and with coefficients in constructible $ \mathbf Q _ {l} $- sheaves, where $ \mathbf Q _ {l} $ is the field of $ l $- adic numbers and where $ l $ is a prime number distinct from the characteristic of the field $ k $. This formula is often called the trace formula.

Let $ X $ be an algebraic variety (or scheme) over a finite field $ k $, let $ F : X \rightarrow X $ be a Frobenius morphism (cf. e.g. Frobenius automorphism), $ {\mathcal F} $ a sheaf on $ X $, and let $ H _ {c} ^ {i} ( X , {\mathcal F} ) $ be cohomology with compact support of the variety (scheme) $ X $ with coefficients in $ {\mathcal F} $. Then the morphism $ F $ determines a cohomology endomorphism

$$ F ^ { * } : H _ {c} ^ {i} ( X , {\mathcal F} ) \rightarrow H _ {c} ^ {i} ( X ,\ {\mathcal F} ) . $$

If $ k _ {n} \supset k $ is an extension of $ k $ of degree $ n $ and if $ X _ {n} = X \otimes k _ {n} $, $ {\mathcal F} _ {n} = {\mathcal F} \otimes k _ {n} $ are the variety (scheme) and sheaf obtained from $ X $ and $ {\mathcal F} $ by extending the field of scalars, then the corresponding Frobenius morphism $ F _ {n} : X _ {n} \rightarrow X _ {n} $ coincides with the $ n $- th power $ F ^ { n } $ of $ F $.

Now let $ X $ be a separable scheme of finite type over the finite field $ k $ of $ q $ elements, let $ {\mathcal F} $ be a constructible $ \mathbf Q _ {l} $- sheaf on $ X $, $ l $ a prime number distinct from the characteristic of $ k $, and $ X ^ {F ^ {n} } $ the set of fixed geometric points of the morphism $ F ^ { n } $ or, equivalently, the set $ X ( k _ {n} ) $ of geometric points of the scheme $ X $ with values in the field $ k _ {n} $. Then for any integer $ n \geq 1 $ the following Lefschetz formula (or trace formula) holds (see [6], [7]):

$$ \tag{2 } \sum _ {x \in X ^ {F ^ {n} } } \mathop{\rm Tr} ( F ^ { n* } , {\mathcal F} _ {x} ) = \ \sum _ { i } (- 1) ^ {i} \mathop{\rm Tr} ( F ^ { * n } , H _ {c} ^ {i} ( X ,\ {\mathcal F} )) , $$

where $ {\mathcal F} _ {x} $ is the stalk of $ {\mathcal F} $ over $ x $. In the case of the constant sheaf $ {\mathcal F} = \mathbf Q _ {l} $ one has $ \mathop{\rm Tr} ( F ^ { n* } , \mathbf Q _ {l} ) = 1 $ and the left-hand side of (2) is none other than the number of geometric points of $ X $ with values in $ k _ {n} $. In particular, for $ n= 1 $ this is simply the number of points of $ X $ with values in the ground field $ k $. If $ X $ is proper over $ k $( for example, if $ X $ is a complete algebraic variety over $ k $), then $ H _ {c} ^ {i} ( X , {\mathcal F} ) = H ^ {i} ( X , {\mathcal F} ) $ and the right-hand side of (2) is an alternating sum of the traces of the Frobenius endomorphism in the ordinary cohomology of $ X $.

There are (see [7]) generalizations of formula (2).

Comments

For the Lefschetz formula in abstract algebraic geometry and its generalizations by A. Grothendieck see also [a1].

References

[1]	S. Lefschetz, "Intersections and transformations of complexes and manifolds" Trans. Amer. Soc. , 28 (1926) pp. 1–49 MR1501331 Zbl 52.0572.02
[2]	S. Lefschetz, "The residual set of a complex manifold and related questions" Proc. Nat. Acad. Sci. USA , 13 (1927) pp. 614–622 Zbl 53.0553.01
[3]	S. Lefschetz, "On the fixed point formula" Ann. of Math. (2) , 38 (1937) pp. 819–822 MR1503373 Zbl 0018.17703 Zbl 63.0563.02
[4]	S.L. Kleiman, "Algebraic cycles and the Weil conjectures" A. Grothendieck (ed.) J. Giraud (ed.) et al. (ed.) , Dix exposés sur la cohomologie des schémas , North-Holland & Masson (1968) pp. 359–386 MR0292838 Zbl 0198.25902
[5]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[6]	P. Deligne, "Cohomologie étale (SGA 4 1/2)" , Lect. notes in math. , 569 , Springer (1977)
[7]	A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , Lect. notes in math. , 589 , Springer (1977) MR491704
[8]	A. Dold, "Lectures on algebraic topology" , Springer (1980) MR0606196 Zbl 0434.55001
[9]	H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) MR0575168 Zbl 0469.55001
[a1]	E. Feitag, R. Kiehl, "Etale cohomology and the Weil conjecture" , Springer (1988) MR926276