Lefschetz trace formula




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



(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.


(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).


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.



(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) \,.


(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.


(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


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.


For ordinary cohomology

The original article is


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.