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.

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

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 $\#\mathrm{Fix}(f)\equiv 2-2g(\mathrm{mod}4)$.