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