On Matveev-Piergallini moves for branched spines

The Matveev-Piergallini (MP) moves on spines of $3$-manifolds are well-known for their correspondence to the Pachner $2$-$3$ moves in dual ideal triangulations. Benedetti and Petronio introduced combinatorial descriptions of closed $3$-manifolds and combed $3$-manifolds by using branched spines and their equivalence relations, which involve MP moves with 16 distinct patterns of branchings. In this paper, we demonstrate that these 16 MP moves on branched spines are derived from a primary MP move, pure sliding moves, and their inverses. Consequently, we obtain simpler combinatorial descriptions for closed $3$-manifolds and combed $3$-manifolds. Furthermore, we extend these results to framed $3$-manifolds and spin $3$-manifolds. These descriptions are advantageous, particularly when constructing and studying quantum invariants of links and $3$-manifolds. In various constructions of quantum invariants using (ideal) triangulations, branching structures naturally arise to facilitate the assignment of non-symmetric algebraic objects to tetrahedra. In these frameworks, the primary MP move precisely corresponds to certain algebraic pentagon relations, such as the pentagon relation of the canonical element of a Heisenberg double, the Biedenharn-Elliott identity for quantum $6j$-symbols, or Schaeffer's identity for the Rogers dilogarithm and its non-commutative analog for Faddeev's quantum dilogarithm in quantum Teichm\"uller theory. We expect our results to contribute to a better understanding of quantum invariants in the context of spines and ideal triangulations.