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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.
 
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 [[#References|[1]]] and for finite cell complexes (see [[#References|[2]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579801.png" />-dimensional sphere into itself. A new version of the proof of the Lefschetz formula for finite cell complexes was given by H. Hopf (see [[#References|[9]]]).
+
This formula was first established by S. Lefschetz for finite-dimensional orientable topological manifolds [[#References|[1]]] and for finite cell complexes (see [[#References|[2]]], [[#References|[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 [[#References|[9]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579802.png" /> be a connected orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579803.png" />-dimensional compact topological [[Manifold|manifold]] or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579804.png" />-dimensional finite [[Cell complex|cell complex]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579805.png" /> be a [[Continuous mapping|continuous mapping]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579806.png" /> be the [[Lefschetz number|Lefschetz number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579807.png" />. Assume that all fixed points of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579808.png" /> are isolated. For each fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l0579809.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798010.png" /> be its Kronecker index (the local degree (cf. [[Degree of a mapping|Degree of a mapping]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798011.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798012.png" />). Then the Lefschetz formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798014.png" /> has the form
+
Let $  X $
 +
be a connected orientable $  n $-
 +
dimensional compact topological [[Manifold|manifold]] or an $  n $-
 +
dimensional finite [[Cell complex|cell complex]], let $  f : X \rightarrow X $
 +
be a [[Continuous mapping|continuous mapping]] and let $  \Lambda ( f , X ) $
 +
be the [[Lefschetz number|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|Degree of a mapping]]) of $  f $
 +
in a neighbourhood of $  x $).  
 +
Then the Lefschetz formula for $  X $
 +
and $  f $
 +
has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\sum _ {f ( x) = x } i ( x)  = \Lambda ( f , X ) .
 +
$$
  
 
There is, [[#References|[8]]], a generalization of the Lefschetz formula to the case of arbitrary continuous mappings of compact Euclidean neighbourhood retracts.
 
There is, [[#References|[8]]], a generalization of the Lefschetz formula to the case of arbitrary continuous mappings of compact Euclidean neighbourhood retracts.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798016.png" /> be a differentiable compact orientable manifold and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798017.png" /> be a differentiable mapping. A fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798019.png" /> is said to be non-singular if it is isolated and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798021.png" /> is the differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798022.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798024.png" /> is the identity transformation. For a non-singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798025.png" /> its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798026.png" /> coincides with the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798027.png" />. In this case the Lefschetz formula (1) shows that the Lefschetz number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798028.png" /> is equal to the difference between the number of fixed points with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798029.png" /> and the number of fixed points with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798030.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798033.png" /> is the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798035.png" /> is the diagonal (cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]).
+
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)|Intersection index (in algebraic geometry)]]).
  
A consequence of the Lefschetz formula is the Hopf formula, which asserts that the [[Euler characteristic|Euler characteristic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798036.png" /> is equal to the sum of the indices of the zeros of a global <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798037.png" />-vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798039.png" /> (it is assumed that all zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798040.png" /> are isolated) (see [[#References|[5]]]).
+
A consequence of the Lefschetz formula is the Hopf formula, which asserts that the [[Euler characteristic|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 [[#References|[5]]]).
  
There is a version of the Lefschetz formula for compact complex manifolds and the Dolbeault cohomology (see [[#References|[5]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798041.png" /> be a compact complex manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798042.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798043.png" /> a be holomorphic mapping with non-singular fixed points. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798044.png" /> be the Dolbeault cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798045.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798046.png" /> (cf. [[Differential form|Differential form]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798047.png" /> be the endomorphism induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798048.png" />. The number
+
There is a version of the Lefschetz formula for compact complex manifolds and the Dolbeault cohomology (see [[#References|[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|Differential form]]) and let $  f ^ { * } : H  ^ {p,q} ( X) \rightarrow H  ^ {p,q} ( X) $
 +
be the endomorphism induced by $  f $.  
 +
The number
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798049.png" /></td> </tr></table>
+
$$
 +
\Lambda ( f , {\mathcal O} _ {X} )  = \sum _ { q= } 0 ^ { m }  (- 1)  ^ {q}  \mathop{\rm Tr} (
 +
f ^ { * } ; H  ^ {0,q} ( X) )
 +
$$
  
 
is called the holomorphic Lefschetz number. One then has the following holomorphic Lefschetz formula:
 
is called the holomorphic Lefschetz number. One then has the following holomorphic Lefschetz formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798050.png" /></td> </tr></table>
+
$$
 +
\Lambda ( f , {\mathcal O} _ {X} )  = \sum _ {f ( x) = x }
 +
\frac{1}{
 +
\mathop{\rm det} ( E - df _ {x} ) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798051.png" /> is the holomorphic differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798052.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798053.png" />.
+
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|Weil cohomology]] in connection with Weil's conjectures about zeta-functions of algebraic varieties defined over finite fields (cf. [[Zeta-function|Zeta-function]]). An analogue of the Lefschetz formula in abstract algebraic geometry has been established for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798054.png" />-adic cohomology with compact support and with coefficients in constructible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798055.png" />-sheaves, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798056.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798057.png" />-adic numbers and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798058.png" /> is a prime number distinct from the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798059.png" />. This formula is often called the trace formula.
+
In abstract algebraic geometry the Lefschetz formula has served as a starting point in the search for [[Weil cohomology|Weil cohomology]] in connection with Weil's conjectures about zeta-functions of algebraic varieties defined over finite fields (cf. [[Zeta-function|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798060.png" /> be an [[Algebraic variety|algebraic variety]] (or [[Scheme|scheme]]) over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798061.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798062.png" /> be a Frobenius morphism (cf. e.g. [[Frobenius automorphism|Frobenius automorphism]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798063.png" /> a sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798064.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798065.png" /> be cohomology with compact support of the variety (scheme) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798066.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798067.png" />. Then the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798068.png" /> determines a cohomology endomorphism
+
Let $  X $
 +
be an [[Algebraic variety|algebraic variety]] (or [[Scheme|scheme]]) over a finite field $  k $,  
 +
let $  F : X \rightarrow X $
 +
be a Frobenius morphism (cf. e.g. [[Frobenius automorphism|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798069.png" /></td> </tr></table>
+
$$
 +
F ^ { * } : H _ {c}  ^ {i} ( X , {\mathcal F} )  \rightarrow  H _ {c}  ^ {i} ( X ,\
 +
{\mathcal F} ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798070.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798071.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798072.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798074.png" /> are the variety (scheme) and sheaf obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798076.png" /> by extending the field of scalars, then the corresponding Frobenius morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798077.png" /> coincides with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798078.png" />-th power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798080.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798081.png" /> be a separable scheme of finite type over the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798082.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798083.png" /> elements, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798084.png" /> be a constructible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798085.png" />-sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798087.png" /> a prime number distinct from the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798088.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798089.png" /> the set of fixed geometric points of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798090.png" /> or, equivalently, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798091.png" /> of geometric points of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798092.png" /> with values in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798093.png" />. Then for any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798094.png" /> the following Lefschetz formula (or trace formula) holds (see [[#References|[6]]], [[#References|[7]]]):
+
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 [[#References|[6]]], [[#References|[7]]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798095.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\sum _ {x \in X ^ {F  ^ {n} } }  \mathop{\rm Tr} ( F ^ { n* } , {\mathcal F} _ {x} )  = \
 +
\sum _ { i } (- 1)  ^ {i}  \mathop{\rm Tr} ( F ^ { * n } , H _ {c}  ^ {i} ( X ,\
 +
{\mathcal F} )) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798096.png" /> is the stalk of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798097.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798098.png" />. In the case of the constant sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l05798099.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980100.png" /> and the left-hand side of (2) is none other than the number of geometric points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980101.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980102.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980103.png" /> this is simply the number of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980104.png" /> with values in the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980105.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980106.png" /> is proper over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980107.png" /> (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980108.png" /> is a complete algebraic variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980109.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980110.png" /> and the right-hand side of (2) is an alternating sum of the traces of the Frobenius endomorphism in the ordinary cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980111.png" />.
+
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 [[#References|[7]]]) generalizations of formula (2).
 
There are (see [[#References|[7]]]) generalizations of formula (2).
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lefschetz,  "Intersections and transformations of complexes and manifolds"  ''Trans. Amer. Soc.'' , '''28'''  (1926)  pp. 1–49</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lefschetz,  "The residual set of a complex manifold and related questions"  ''Proc. Nat. Acad. Sci. USA'' , '''13'''  (1927)  pp. 614–622</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lefschetz,  "On the fixed point formula"  ''Ann. of Math. (2)'' , '''38'''  (1937)  pp. 819–822</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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 &amp; Masson  (1968)  pp. 359–386</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Griffiths,  J.E. Harris,  "Principles of algebraic geometry" , Wiley (Interscience)  (1978)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Deligne,  "Cohomologie étale (SGA 4 1/2)" , ''Lect. notes in math.'' , '''569''' , Springer  (1977)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A. Grothendieck,  I. Bucur,  C. Honzel,  L. Illusie,  J.-P. Jouanolou,  J.-P. Serre,  "Cohomologie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980112.png" />-adique et fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057980/l057980113.png" />. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer  (1977)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Dold,  "Lectures on algebraic topology" , Springer  (1980)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Seifert,  W. Threlfall,  "A textbook of topology" , Acad. Press  (1980)  (Translated from German)</TD></TR></table>
 
 
 
  
 
====Comments====
 
====Comments====
Line 48: Line 175:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Feitag,   R. Kiehl,   "Etale cohomology and the Weil conjecture" , Springer (1988)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lefschetz, "Intersections and transformations of complexes and manifolds" ''Trans. Amer. Soc.'' , '''28''' (1926) pp. 1–49 {{MR|1501331}} {{ZBL|52.0572.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lefschetz, "The residual set of a complex manifold and related questions" ''Proc. Nat. Acad. Sci. USA'' , '''13''' (1927) pp. 614–622 {{MR|}} {{ZBL|53.0553.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lefschetz, "On the fixed point formula" ''Ann. of Math. (2)'' , '''38''' (1937) pp. 819–822 {{MR|1503373}} {{ZBL|0018.17703}} {{ZBL|63.0563.02}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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 &amp; Masson (1968) pp. 359–386 {{MR|0292838}} {{ZBL|0198.25902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Deligne, "Cohomologie étale (SGA 4 1/2)" , ''Lect. notes in math.'' , '''569''' , Springer (1977)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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) {{MR|491704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German) {{MR|0575168}} {{ZBL|0469.55001}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Feitag, R. Kiehl, "Etale cohomology and the Weil conjecture" , Springer (1988) {{MR|926276}} {{ZBL|}} </TD></TR></table>

Latest revision as of 19:04, 26 March 2023


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
How to Cite This Entry:
Lefschetz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lefschetz_formula&oldid=16459
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article