Lefschetz number of a finite CW self-map

Let $X$ be a finite CW complex and $f:X\to X$ a continuous map. Its Lefschetz number is $L(f)={\sum }_{i\ge 0}(-1{)}^i\mathrm{tr}(f_{\ast }:H_i(X;\mathbb{Q})\to H_i(X;\mathbb{Q}))\in \mathbb{Q}\mathrm{.}$ Trace means the basis-independent trace of The basis-independent trace of an endomorphism of a finite-dimensional vector space. The induced maps are rational-linear because postcomposition on singular chains preserves the rational coefficients.

This sum is well-defined and finite. By Relative homology of consecutive CW skeleta, the cellular chain group in degree $i$ is one copy of $\mathbb{Q}$ for each $i$-cell. Since there are finitely many cells, these are finite-dimensional and zero above the largest cell dimension. Their cycle subspaces and boundary subspaces are finite-dimensional, and so are their quotients: in a vector space with a finite basis, successively choosing an independent vector in a subspace can take at most the ambient dimension steps, since elimination of coordinates bounds the length of any independent list by that number. A maximal such finite list spans the subspace. Extending a basis of the boundary subspace to one of the cycle subspace gives a finite spanning basis of their quotient. These are only finite choices. The comparison in Cellular homology computes singular homology therefore makes each $H_i(X;\mathbb{Q})$ finite-dimensional and zero above that largest dimension, as required. This use is for the dimensions of the groups; $f$ need not be cellular.

Homotopic maps have the same induced homology endomorphisms by Homotopic maps induce the same map on singular homology, hence the same Lefschetz number. No extra choice principle is used in this definition or its well-definedness.

For empty $X$, all groups vanish and $L(f)=0$, the empty sum. For a point, only $H_0=\mathbb{Q}$ is nonzero and the only self-map has Lefschetz number $1$. For any finite discrete $X$, the matrix on the point basis of $H_0$ has diagonal entry $1$ exactly at a fixed point and $0$ otherwise, so $L(f)$ counts its fixed points. Zero-dimensional and zero vector-space traces therefore obey the same formula. Higher degenerate singular simplices are included in cellular comparison and do not create extra homology terms.