Lefschetz fixed-point theorem

If $L(f)\ne 0$, then $f$ has a fixed point. For a fixed-point-free map, sufficiently fine simplicial approximation has zero diagonal chain coefficients, hence zero Lefschetz number.

A closed smooth manifold X embeds in Euclidean space. The double M of a closed tubular neighbourhood is a closed oriented manifold retracting onto X. If i is its inclusion and r its retraction, the fixed points of F correspond to those of f, and cyclicity of finite-dimensional trace gives $L(F)=L(f)$ from $i^{\ast }{r}^{\ast }=1$. Thus the oriented diagonal-class proof of the Lefschetz fixed-point theorem implies the theorem for possibly nonorientable X, without incorrectly treating its diagonal as cooriented.

Double a compact orientable genus-zero topological surface with $g+1$ boundary circles. The resulting closed surface has genus $g$. The involution exchanging the halves fixes the seam circles and has local collar form $(s,t)\mapsto (s,-t)$. Compose it with a flow rotating each seam coordinate by a small nonzero amount, with speed supported in its collar. Equality with the initial point would force $t=0$, where the rotation is nonzero; off the collars the original reflection already has no fixed points. The new map is fixed-point-free and homotopic to the reflection through the flow. This gives a surface example in every genus where having fixed points is not a homotopy invariant.

Every self-map has $f^{\ast }a=\lambda a$ in the mod-two cohomology ring, with $\lambda \in {\mathbb{F}}_2$. Its mod-two Lefschetz number is $1+\lambda +\cdots +{\lambda }^{2r}$, equal to one for either value of $\lambda$. This is the reduction of the integral cellular Lefschetz number, so the latter is nonzero and a fixed point exists.

Such a self-map of ${\Sigma }_{g_1}\times {\Sigma }_{g_2}$ exists exactly when one factor is a torus. Necessity follows because its Lefschetz number equals the Euler characteristic $4(1-g_1)(1-g_2)$ and must vanish. Sufficiency is translation by a nonzero point in the torus factor, times the identity on the other factor. The condition does not determine the product $g_1{g}_2$, since the other genus can be arbitrary.

Let $M$ be a closed oriented $n$-manifold, let $e_{p,i}$ be a homogeneous basis of $H^p(M;\mathbb{Q})$, and choose its Poincare-dual basis $e_{p,i}^{\ast }\in H^{n-p}(M;\mathbb{Q})$ by $⟨e_{p,i}{e}_{p,j}^{\ast },[M]⟩={\delta }_{ij}$. With the product-orientation convention, the Poincare dual of the diagonal is

The cohomology ring of the product of two real projective planes is ${\mathbb{F}}_2[x,y]/(x^3,y^3)$, with $|x|=|y|=1$. Pairing against $x^2$, $xy$ and $y^2$ identifies the Poincare dual of the diagonal as $x^2+xy+y^2$. For a self-map $f$ with $f^{\ast }x=\epsilon x$, the graph pullback evaluates on the mod-two fundamental class as $1+\epsilon +{\epsilon }^2=1$. Thus no graph of a map on the whole projective plane can avoid the diagonal. The argument uses a relative Thom class and does not assume transverse intersections.

For the graph map $(1,f):M\to M\times M$, the cohomology class of the diagonal satisfiesIf $f$ is fixed-point-free, its graph misses the diagonal, so the pullback vanishes. This proves the Lefschetz fixed-point theorem for closed oriented manifolds.

Let $F$ be the self-map of the double $D(N)=N{\cup }_{\partial N}N$ induced by a self-map $f$ preserving the boundary. Naturality of the Mayer–Vietoris sequence and alternating-trace additivity give

Let $f$ have prime order $p$ on a finite CW complex, and suppose every nonidentity power is fixed-point-free. On rational homology,is the projection onto the invariant subspace. ThereforeThe left side is an integer, so $p$ divides $\chi (X)$.

Take $n$ labelled copies of $S^2$ and identify one marked point of copy $i$ with the antipodal marked point of copy $i+1$ cyclically. The resulting connected finite CW complex is homotopy equivalent toso its rational homology has dimensions $1,1,n$ in degrees zero, one, and two, and its Euler characteristic is $n$.

If a smooth self-map of a compact manifold has only isolated nondegenerate fixed points, thenEvery summand has absolute value one, so the number of fixed points is at least $|L(f)|$.

Let $W_g$ be a connected sum of oriented manifolds consisting of $g$ copies of $S^r\times S^r$ with odd $r\ge 1$. Suppose an orientation-preserving smooth involution $f$ has finitely many fixed points, all nondegenerate with positive determinant $\det (I-D{f}_x)$. The Lefschetz-Hopf fixed-point theorem and the cohomology ring of a connected sum of odd-dimensional sphere products giveThis middle group has dimension $2g$ and its Poincare duality pairing is alternating. The induced map preserves the pairing and squares to the identity. By eigenspaces of a symplectic involution, its trace is $2g-4b$ for an integer $b$. Therefore $\#\mathrm{Fix}(f)\equiv 2-2g(\mathrm{mod}4)$.