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For a finite simplicial complex $K$, let $f_i$ count its $i$-dimensional simplexes. Its Euler characteristic iswhere $b_i={\dim }_{\mathbb{Q}}{H}_i(|K|;\mathbb{Q})$ is its rational Betti number.

Work over $\mathbb{Q}$, so the chain groups, cycle modules and boundary modules are finite-dimensional vector spaces. Choosing vector-space complements gives $C_i\cong B_i\oplus H_i\oplus B_{i-1}$. The boundary dimensions occur twice in the alternating sum, with opposite signs, and cancel. This proves the formula; it does not require splitting off possibly torsion integral homology as a direct summand of an integral cycle module. The rank cancellation in the Euler-Poincare formula gives the same argument without choosing complements.

Let $X$ be the geometric realization of a finite simplicial complex, or a retract of such a space, and let $f:X\to X$ be continuous. If its Lefschetz number $L(f)$ is nonzero, then $f$ has a fixed point. This includes self-maps of compact manifolds.^[ref]

For a map without fixed points on a finite polyhedron, compactness gives a positive separation between $x$ and $f(x)$. After sufficiently fine simplicial subdivision, a simplicial approximation sends no simplex back onto itself. Its diagonal chain map coefficients vanish, so its alternating trace, and hence $L(f)$, is zero. A retraction reduces the more general stated case to this one.

For example, every continuous self-map of an even-dimensional sphere with mapping degree different from $-1$ has a fixed point, since $L(f)=1+\mathrm{deg}f$. The converse fails: $L(f)=0$ permits fixed points, as the identity map of a circle shows.

For a continuous self-map $f:X\to X$, suppose the rational homology groups $H_i(X;\mathbb{Q})$ are finite-dimensional and vanish in sufficiently high degrees. Its Lefschetz number is the finite alternating traceFor a self-map of a finite simplicial complex, it also equals the alternating trace on any finite simplicial chain complex representing the induced map. The traces on boundaries cancel in consecutive degrees, leaving precisely the traces on homology. Thus a potentially large chain computation reduces to the homology groups.

The identity map has $L(1_X)=\chi (X)$, the Euler characteristic. On a circle, a map of mapping degree $d$ has $L(f)=1-d$. A nonzero value forces a fixed point under the hypotheses of the Lefschetz fixed-point theorem; finiteness of the displayed sum alone is not that theorem's full geometric hypothesis.

For a geometric simplicial complex $K$ in a Euclidean space, its mesh is the supremum of its simplex diameters:If $K$ has finite dimension $n$, its barycentric subdivision satisfiesThe vertices of a subdivided simplex are barycentres of nested faces. Bounding the distances between those barycentres gives the displayed factor, uniformly over all original simplexes.

Consequently, if the initial mesh is finite, iterating gives $\mu ({\mathrm{sd}}^{r}K)\le (n/(n+1){)}^r\mu (K)\to 0$. In particular this holds for a finite simplicial complex. An infinite geometric complex with unbounded edge lengths can instead have infinite mesh after every finite subdivision, so finite dimension alone does not imply the asserted convergence.