Lefschetz fixed-point theorem
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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the number of fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.
The counting is subject to an imputed multiplicity at a fixed point called the fixed point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).
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[edit] Formal statement
For a formal statement of the theorem, let
- Failed to parse (Cannot write to or create math output directory): f: X \rightarrow X\,
be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λf of f by
- Failed to parse (Cannot write to or create math output directory): \Lambda_f:=\sum_{k\geq 0}(-1)^k\mathrm{Tr}(f_*|H_k(X,\mathbb{Q})),
the alternating (finite) sum of the matrix traces of the linear maps induced by f on the Hk(X,Q), the singular homology of X with rational coefficients.
A simple version of the Lefschetz fixed-point theorem states: if
- Failed to parse (Cannot write to or create math output directory): \Lambda_f \neq 0\,
then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.
Note however that the converse is not true in general: Λf may be zero even if f has fixed points.
A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if f has only finitely many fixed points, then
- Failed to parse (Cannot write to or create math output directory): \sum_{x \in \mathrm{Fix}(f)} i(f,x) = \Lambda_f,
where Fix(f) is the set of fixed points of f, and i(f,x) denotes the index of the fixed point x.
[edit] Relation to the Euler characteristic
The Lefschetz number of the identity map on a finite CW complex can be easily computed by realizing that each Failed to parse (Cannot write to or create math output directory): \scriptstyle f_\ast
can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic χ(X). Thus we have
- Failed to parse (Cannot write to or create math output directory): \Lambda_{\mathrm{id}} = \chi(X).\
[edit] Relation to the Brouwer fixed point theorem
The Lefschetz fixed point theorem generalizes the Brouwer fixed point theorem, which states that every continuous map from the n-dimensional closed unit disk Dn to Dn must have at least one fixed point.
This can be seen as follows: Dn is compact and triangulable, all its homology groups except H0 are 0, and every continuous map f : Dn → Dn induces a non-zero homomorphism f* : H0(Dn, Q) → H0(Dn, Q); all this together implies that Λf is non-zero for any continuous map f : Dn → Dn.
[edit] Historical context
Lefschetz presented his fixed point theorem in [Lefschetz 1926]. Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence points of mappings.
Given two maps f and g from a manifold X to a manifold Y, the Lefschetz coincidence number of f and g is defined as
- Failed to parse (Cannot write to or create math output directory): \Lambda_{f,g} = \sum (-1)^k \mathrm{Tr}( D_X \circ g^* \circ D_Y^{-1} \circ f_*),
where f∗ is as above, g∗ is the mapping induced by g on the cohomology groups with rational number coefficients, and DX and DY are the Poincaré duality isomorphisms for X and Y, respectively.
Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X = Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.
[edit] Frobenius
Let Failed to parse (Cannot write to or create math output directory): X\,
be a variety defined over the finite field Failed to parse (Cannot write to or create math output directory): k with Failed to parse (Cannot write to or create math output directory): q elements and let Failed to parse (Cannot write to or create math output directory): \bar X be the lift of Failed to parse (Cannot write to or create math output directory): X\, to the algebraic closure of Failed to parse (Cannot write to or create math output directory): k
. The Frobenius endomorphism (often just the Frobenius), notation Failed to parse (Cannot write to or create math output directory): F_q
, of Failed to parse (Cannot write to or create math output directory): \bar X maps a point with coordinates Failed to parse (Cannot write to or create math output directory): x_1,\ldots,x_n to the point with coordinates Failed to parse (Cannot write to or create math output directory): x_1^q,\ldots,x_n^q
. Thus the fixed points of Failed to parse (Cannot write to or create math output directory): F_q
are exactly the points of Failed to parse (Cannot write to or create math output directory): X with coordinates in Failed to parse (Cannot write to or create math output directory): k
, notation for the set of these points: Failed to parse (Cannot write to or create math output directory): X(k) . The Lefschetz trace formula holds in this context and reads:
- Failed to parse (Cannot write to or create math output directory): \#X(k)=\sum_i (-1)^i \mathop{\rm tr} F_q| H^i_c(\bar X,{\Bbb Q}_\ell).
This formula involves the trace of the Frobenius on the étale cohomology with compacts supports of Failed to parse (Cannot write to or create math output directory): \bar X
with values in the field of Failed to parse (Cannot write to or create math output directory): \ell
-adic numbers, where Failed to parse (Cannot write to or create math output directory): \ell
is a prime prime to Failed to parse (Cannot write to or create math output directory): q
.
If Failed to parse (Cannot write to or create math output directory): X
is smooth, this formula can be rewritten in terms of the arithmetic Frobenius Failed to parse (Cannot write to or create math output directory): \Phi_q
, which acts as the inverse of Failed to parse (Cannot write to or create math output directory): F_q
on cohomology:
- Failed to parse (Cannot write to or create math output directory): \#X(k)=q^{\dim X}\sum_i (-1)^i \mathop{\rm tr} \Phi_q| H^i(\bar X,{\Bbb Q}_\ell).
This formula involves usual cohomology, not cohomology with compact supports.
[edit] See also
[edit] References
- Solomon Lefschetz (1926). "Intersections and transformations of complexes and manifolds". Trans. Amer. Math. Soc. 28: 1–49. doi:. jstor
- Solomon Lefschetz (1937). "On the fixed point formula". Ann. of Math.(4) 38: 819–822. doi:.
