 Lefschetz number

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.