Theorem. The abelianized total orientation [frap-000S]
✍️source

    Let \(I\) be finite, let \(K\) act faithfully on a nonempty \(\Omega \), and choose bijective frames \(c_i:\Omega \to X_i\). For a framed transport \(g=(\pi ,a)\), let \(K_{\mathrm {ab}}=K/[K,K]\) be its abelianization, where \([K,K]\) is generated by \(uvu^{-1}v^{-1}\), and write \([a]\) for the class of \(a\). The total-orientation character

    \[\chi (g)=\prod _{i\in I}[a_i]\quad \hbox {in }K_{\mathrm {ab}}\]

    is a homomorphism, independent of replacing the frames by \(c_i'=c_i t_i\) for \(t_i\in K\). This is the canonical character in [marchetti2025rubik, Section 3.1, p. 7]; the proof below gives both assertions directly in source-indexed coordinates.

    Proof. 

      For \(h=(\sigma ,b)\), the composition law and commutativity of the target give

      \[\chi (hg)=\prod _i[b_{\pi (i)}][a_i] =\left (\prod _j[b_j]\right )\left (\prod _i[a_i]\right ) =\chi (h)\chi (g).\]

      The middle equality reindexes using the bijection \(\pi \). In the new frames the local map is \(a_i'=t_{\pi (i)}^{-1}a_i t_i\). Its abelianized product contains each \([t_i]\) and its inverse exactly once, so these cancel and leave \(\chi (g)\).

      Frame independence here concerns changes within the chosen group of frames. Reversing a common orientation convention can also change the identification of the target group. The target and its quotient map are Abelianization and Abelianization.of; the total product and its frame independence are the argument above.