The two lifts of a self-map carry twice the fixed point index sum

Fixed points over a fixed point. Let $x\in \mathrm{Fix}(f)$ and ${\pi }^{-1}(x)=\{a,\tau a\}$. Since $\pi (\tilde{f}(a))=f(\pi (a))=f(x)=x$, the point $\tilde{f}(a)$ lies in $\{a,\tau a\}$. If $\tilde{f}(a)=a$, then $\tilde{f}(\tau a)=\tau \tilde{f}(a)=\tau a$ by the commutation, so both fibre points are fixed by $\tilde{f}$, while ${\tilde{f}}^{\mathrm{\prime }}(a)=\tau (a)=\tau a\mathrm{\ne }a$ and ${\tilde{f}}^{\mathrm{\prime }}(\tau a)=a\mathrm{\ne }\tau a$; if $\tilde{f}(a)=\tau a$, then $\tilde{f}(\tau a)=\tau \tilde{f}(a)={\tau }^{2}a=a$, so both fibre points are fixed by ${\tilde{f}}^{\mathrm{\prime }}=\tau \tilde{f}$ and neither by $\tilde{f}$. In both cases exactly two of the four pairs $(\tilde{g},\tilde{x})\in \{\tilde{f},{\tilde{f}}^{\mathrm{\prime }}\}\times {\pi }^{-1}(x)$ satisfy $\tilde{g}(\tilde{x})=\tilde{x}$, namely one lift fixing both points of the fibre. Conversely, a fixed point of $\tilde{f}$ or of ${\tilde{f}}^{\mathrm{\prime }}$ projects to a fixed point of $f$, because $\pi \circ \tilde{f}=f\circ \pi$.

Isolation and finiteness. Let $\tilde{x}$ be a fixed point of $\tilde{f}$ and $x=\pi (\tilde{x})$. Choose a neighbourhood $W$ of $x$ containing no fixed point of $f$ other than $x$. Since $\pi$ is a local homeomorphism and $\tilde{M}$ is Hausdorff, choose a neighbourhood $\tilde{W}$ of $\tilde{x}$ with $\pi (\tilde{W})\subseteq W$ and $\tau \tilde{x}\mathrm{\notin }\tilde{W}$. Any fixed point of $\tilde{f}$ in $\tilde{W}$ projects into $W$, hence lies over $x$, hence is $\tilde{x}$ or $\tau \tilde{x}$; the second is excluded by $\tau \tilde{x}\mathrm{\notin }\tilde{W}$. So the fixed points of $\tilde{f}$ are isolated, and the same argument applies to ${\tilde{f}}^{\mathrm{\prime }}$. Their fixed sets are closed by [L1] and discrete, and $\tilde{M}$ is compact by [F1], so both fixed sets are finite by [L1].

Local indices. Let $\tilde{x}$ be a fixed point of $\tilde{f}$ with $\pi (\tilde{x})=x$. Since $\pi$ is a local diffeomorphism, it restricts to a diffeomorphism from an open neighbourhood of $\tilde{x}$ onto an open neighbourhood of $x$, and from $\pi \circ \tilde{f}=f\circ \pi$ we get $\tilde{f}={\pi }^{-1}\circ f\circ \pi$ on that neighbourhood; the conjugation lemma [F2] therefore gives ${\mathrm{ind}}_{\tilde{x}}(\tilde{f})={\mathrm{ind}}_x(f)$. The same computation applies to every fixed point of ${\tilde{f}}^{\mathrm{\prime }}$, which also satisfies $\pi \circ {\tilde{f}}^{\mathrm{\prime }}=f\circ \pi$. Step 1.1 says that over each fixed point $x$ of $f$ exactly one of the two lifts has fixed points, and it fixes both points of the fibre, so the total of the local indices of $\tilde{f}$ and ${\tilde{f}}^{\mathrm{\prime }}$ over $x$ is $2{\mathrm{ind}}_x(f)$. Summing over the finite set $\mathrm{Fix}(f)$ — the geometric Lefschetz number of $f$ is the finite index sum of Geometric Lefschetz number (index sum) — gives the displayed identity.