Proposition. Transfer computes total local orientation [frap-000X]
✍️source

    Let \(G\) act transitively on a finite nonempty position set \(I\), and on a tagged union of fibers \(X_i\), carrying \(X_i\) bijectively to \(X_{g i}\). Fix \(i_0\in I\) and put \(H=G_{i_0}\). Let \(K\) act faithfully on a nonempty set \(\Omega \); choose a bijective base frame \(c_0:\Omega \to X_{i_0}\) and a homomorphism \(\rho :H\to K\) satisfying \(h c_0=c_0\rho (h)\) for every \(h\in H\). Choose \(r_i\in G\) with \(r_i i_0=i\), and set \(c_i=r_i c_0\). Then the total-orientation character of this framed transport is

    \[\chi =\rho _{\mathrm {ab}}\circ T_{G,H}:G\longrightarrow K_{\mathrm {ab}},\]

    where \(T_{G,H}\) is transfer and \(\rho _{\mathrm {ab}}([h])=[\rho (h)]\). This identifies the coset construction in [mayer2015artin, Section 3.3, p. 6] with the local-frame coordinates of [marchetti2025rubik, Section 3.1, pp. 6--7].

    Proof. 

      The map \(G/H\to I\), \(rH\mapsto r i_0\), is a bijection: \(r i_0=s i_0\) exactly when \(s^{-1}r\in H\). Thus \(H\) has finite index \(|I|\), and the \(r_i\) are left-coset representatives. Write \(\pi _g(i)=g i\) and \(g r_i=r_{\pi _g(i)}h_i(g)\). Then

      \[g c_i=g r_i c_0=r_{\pi _g(i)}h_i(g)c_0 =c_{\pi _g(i)}\rho (h_i(g)).\]

      Consequently every local factor belongs to \(K\), and its source-indexed coordinate is \(a_i(g)=\rho (h_i(g))\). A homomorphism sends commutators to commutators, so \(\rho _{\mathrm {ab}}\) is well defined. Multiplying the local classes gives

      \[\chi (g)=\prod _i[\rho (h_i(g))] =\rho _{\mathrm {ab}}\left (\prod _i[h_i(g)]\right ).\]

      The vertical arrows are the transported frames. The equality \(g r_i=r_{\pi _g(i)}h_i(g)\) makes the square commute: after the destination representative is removed, the remaining motion fixes \(i_0\) and acts through \(\rho \). Changing representatives changes these frames within \(K\), so the character is unchanged.

      The induced map is Abelianization.map. Naturality of transfer is MonoidHom.transfer_comp, with \(\phi :H\to H_{\mathrm {ab}}\) the quotient map and the target map \(\rho _{\mathrm {ab}}\). That algebraic identity supports the last equality; the identification with the actual fiber action is the transported-frame argument above.