The Lefschetz numbers of the two lifts sum to twice the base Lefschetz number

The transfer. For a singular simplex $\sigma$ let $T_{\mathrm{\#}}(\sigma )$ be the sum of its two lifts through $\pi$; the sum over the full lift set is independent of any selection, so it defines a rational-linear map $T_{\mathrm{\#}}:C_{\ast }(M;\mathbb{Q})\to C_{\ast }(\tilde{M};\mathbb{Q})$. It is a chain map: restriction to a face bijects the lift set of $\sigma$ with the lift set of its faces, since a lift of a face extends uniquely along the inclusion of the connected, simply connected simplex, so boundaries commute with $T_{\mathrm{\#}}$. By [F1] and [F2], ${\pi }_{\mathrm{\#}}{T}_{\mathrm{\#}}=2\mathrm{i}\mathrm{d}$ and $T_{\mathrm{\#}}{\pi }_{\mathrm{\#}}=\mathrm{i}\mathrm{d}+{\tau }_{\mathrm{\#}}$ on chains, hence on homology ${\pi }_{\ast }{T}_{\ast }=2\mathrm{i}\mathrm{d}$ and $T_{\ast }{\pi }_{\ast }=\mathrm{i}\mathrm{d}+{\tau }_{\ast }$. Also ${\tau }_{\ast }{T}_{\ast }=T_{\ast }$ because the deck involution exchanges the two summands of every transfer.

The invariant decomposition. Since ${\tau }^2=\mathrm{i}\mathrm{d}$, the involution ${\tau }_{\ast }$ of $H_{\ast }(\tilde{M};\mathbb{Q})$ has eigenvalues $\pm 1$ and splits the space as $V_{+}\oplus V_{-}$. From step 1.1, $T_{\ast }(y)\mathrm{/}2\in V_{+}$ and ${\pi }_{\ast }(T_{\ast }(y)\mathrm{/}2)=y$ for every $y$, so ${\pi }_{\ast }{\mathrm{\mid }}_{V_{+}}$ is surjective; and ${\pi }_{\ast }{\mathrm{\mid }}_{V_{+}}$ is injective, because ${\pi }_{\ast }x=0$ with $x\in V_{+}$ gives $0=T_{\ast }{\pi }_{\ast }x=x+{\tau }_{\ast }x=2x$; hence ${\pi }_{\ast }$ restricts to an isomorphism $V_{+}\to H_{\ast }(M;\mathbb{Q})$. Naturality ${\pi }_{\ast }{\tilde{f}}_{\ast }=f_{\ast }{\pi }_{\ast }$ conjugates ${\tilde{f}}_{\ast }{\mathrm{\mid }}_{V_{+}}$ to $f_{\ast }$, and because $\tilde{f}$ commutes with $\tau$ the map ${\tilde{f}}_{\ast }$ preserves each $V_{\pm }$.

The trace identities. By step 2.1, $\mathrm{tr}({\tilde{f}}_{\ast }\mathrm{\mid }{V}_{+}^i)=\mathrm{tr}(f_{\ast }\mathrm{\mid }{H}_i(M;\mathbb{Q}))$, and $(\tau \tilde{f}{)}_{\ast }={\tau }_{\ast }{\tilde{f}}_{\ast }$ acts as $+{\tilde{f}}_{\ast }$ on $V_{+}$ and $-{\tilde{f}}_{\ast }$ on $V_{-}$; hence the alternating sums satisfy $L(\tilde{f})={\sum }_i(-1{)}^i[\mathrm{tr}({\tilde{f}}_{\ast }\mathrm{\mid }{V}_{+}^i)+\mathrm{tr}({\tilde{f}}_{\ast }\mathrm{\mid }{V}_{-}^i)]$ and $L(\tau \tilde{f})={\sum }_i(-1{)}^i[\mathrm{tr}({\tilde{f}}_{\ast }\mathrm{\mid }{V}_{+}^i)-\mathrm{tr}({\tilde{f}}_{\ast }\mathrm{\mid }{V}_{-}^i)]$, whose sum is $2{\sum }_i(-1{)}^i\mathrm{tr}(f_{\ast }\mathrm{\mid }{H}_i(M;\mathbb{Q}))=2L(f)$. The same computation may be read in cohomology by [L1], which is how the source states the transfer. AC enters only through the finiteness in Algebraic Lefschetz number via rational homology traces; the transfer itself is canonical and the two-lift sums involve no selection.