Lefschetz number

Remarks

Homotopic maps induce the same maps on homology, so $L(f)$ depends only on the homotopy class of $f$. For the identity map, $L(\id_X) = \sum_k (-1)^k \dim H_k(X; \Q) = \chi(X)$, the Euler characteristic. One may use $H_k(X; \Z)$ modulo torsion instead of rational homology and get the same number.

Example: let $X$ be connected and let $f$ be constant. Then $f_{\ast,0}$ is the identity of $H_0(X; \Q) \cong \Q$ and $f_{\ast,k} = 0$ for $k > 0$, so $L(f) = 1$.

Example: the antipodal map $a$ of $S^n$ acts on $H_n(S^n; \Q)$ by its degree $(-1)^{n+1}$, so $L(a) = 1 + (-1)^n (-1)^{n+1} = 0$, in line with the fact that $a$ has no fixed point.

Non-example: for an infinite wedge of circles, $H_1(X; \Q)$ is infinite-dimensional, so the trace in the formula is not defined; the finiteness hypothesis on $X$ fails.