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Lefschetz trace formula

Contents

Contents

Idea

A formula for the number of fixed points of continuous endo-maps of topological spaces.

Statement

Definition

(Lefschetz number)
Given a continuous endomorphism f:XXf \colon X\to X of a topological space, its Lefschetz number is the alternating sum of the traces

Λ k(X,f) i(1) iTr(H i(X,)f *H i(X,)), \Lambda_k(X,f) \;\coloneqq\; \sum_i \, (-1)^i \cdot Tr \bigg( H^i(X ,\, \mathbb{Q}) \xrightarrow{ f^\ast } H^i(X ,\, \mathbb{Q}) \bigg) \,,

of the linear endomorphisms on the rational cohomology groups with coefficients given by pullback in cohomology .

One sometimes also speaks of the Lefschetz number of the induced endomorphism of the chain/cochain complexes, see algebraic Lefschetz formula.

Proposition

(Lefschetz fixed point theorem)
If XX is a compact polyhedron (a finite simplicial complex) and if the Lefschetz number (Def. ) is non-zero, then ff has at least one fixed point. I

The proof follows from the existence of

  1. a good cycle map,

  2. the Künneth formula,

  3. Poincaré duality.

(see Milne, section 25).

Remark

The Lefschetz formula holds more generally in Weil cohomology theories (by definition) and hence notably in ℓ-adic étale cohomology. This fact serves to prove the Weil conjectures.

Examples

Example

(Euler characteristic)
For f=idf = id is the identity map, the Lefschetz number (Def. ) reduces to the Euler characteristic of XX (see this Def.):

Λ(X,id)=χ(X). \Lambda(X,\,id) \,=\, \chi(X) \,.

Example

(fixed point theorem for contractible polyhedra)
If XX is compact polyhedron which is contractible then all its cohomology groups in positive degree vanish, H 1(X;k)=0H^{\geq 1}(X; k) \,=\, 0, while H 0(X;k)=kH^0(X; k) \,=\, k. This implies that the Lefschetz number (Def. ) of every endo-map ff is Λ(X,f)=1\Lambda(X,f) \,=\, 1.

Therefore, for contactible compact polyhedral XX the Lefschetz fixed point theorem (Prop. ) says that every map f:XXf \colon X\to X has a fixed point.

In the further special case that X=D nX = D^n is a disk/ball, this is also the statement of Brouwer's fixed point theorem.

Example

(fixed point theorem for homeomorphisms of n-spheres)
For nn \in \mathbb{N}, the n-sphere has ordinary cohomology, in particular, rational cohomology, concentrated in degrees 0 and nn:

H k(S n;){ | k{0,n} 0 | otherwise. H^k(S^n;\, \mathbb{Q}) \;\simeq\; \left\{ \array{ \mathbb{Q} &\vert& k \in \{0,n\} \\ 0 &\vert& \text{otherwise} } \right. \,.

Any map f:S nS nf \,\colon\, S^n \to S^n necessarily induces the identity on H 0(S n;)H^0(S^n;\mathbb{Q}) and is multiplication by the degree d(f)d(f) on H n(S n;)H^n(S^n; \mathbb{Q}).

Therefore the Lefschetz number of ff (Def. ) is

Λ(S n,f)=1+(1) nd(f) \Lambda(S^n, f) \;=\; 1 + (-1)^n d(f)

and so the Lefschetz fixed point theorem (Prop. ) in this case implies that sufficient conditions for ff to have a fixed point is that

  • ff is not a homeomorphism (its degree is different from ±1\pm 1);

  • ff is a homeomorphism and

    d(f)=(1) n. d(f) \,=\, (-1)^n \,.

The last case means equivalently that a homeomorphism f:S nS nf \,\colon\, S^n \to S^n is guaranteed to have a fixed point if

or

This plays a role in the classification of free transformation group actions on n-spheres, see at group actions on spheres the section Free actions by finite groups.

For example, the antipodal reflection action of / 2 \mathbb{Z}/2 on S nS^n is orientation reversing for even nn and orientation preserving for odd nn and hence always has vanishing Lefschetz number. Indeed, this action is manifestly fixed-point free. But the Lefschetz fixed point theorem implies that this is the only case, in that there can be no orientation-preserving free action of /2\mathbb{Z}/2 on an even-dimensional sphere, and no orientation-reversing free action of /2\mathbb{Z}/2 on an odd-dimensional sphere.

References

For ordinary cohomology

The original article is

Review:

In equivariant cohomology:

For étale cohomology

For étale cohomology of schemes:

For algebraic stacks:

Last revised on October 27, 2021 at 09:41:20. See the history of this page for a list of all contributions to it.