The Lefschetz number of the identity is the Euler characteristic

The identity's traces. By [F1] the induced map $({\mathrm{i}\mathrm{d}}_M{)}_{\ast }$ is the identity of $H_i(M;\mathbb{Q})$ for each $i$, so $\mathrm{tr}(({\mathrm{i}\mathrm{d}}_M{)}_{\ast })={\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})$; therefore $L({\mathrm{i}\mathrm{d}}_M)={\sum }_i(-1{)}^i{\dim }_{\mathbb{Q}}{H}_i(M;\mathbb{Q})$, the same finite alternating sum that defines $\chi (M)$ in [F2]. Hence $L({\mathrm{i}\mathrm{d}}_M)=\chi (M)$.

Maps homotopic to the identity. If $f$ is smooth and homotopic to ${\mathrm{i}\mathrm{d}}_M$, then $L(f)=L({\mathrm{i}\mathrm{d}}_M)=\chi (M)$ by [L1] and step 1.1. When in addition $\dim M\ge 1$ and $f$ has isolated fixed points, the same number is the geometric index sum $I(f)$, which is how the identity's Lefschetz number is recovered geometrically by a small perturbation of the identity. For nonempty $M$ with $n\ge 1$, every point is fixed by ${\mathrm{i}\mathrm{d}}_M$ and no point is isolated, so its geometric index sum is not defined directly. If $M=\mathrm{\varnothing }$ and $n\ge 1$, the identity has no fixed points and $I({\mathrm{i}\mathrm{d}}_M)=0=L({\mathrm{i}\mathrm{d}}_M)=\chi (M)$.