Sets of moves for standard skeleta of 3-manifolds with marked boundary

A 3-manifold with marked boundary is a pair (M, X), where M is a compact 3-manifold whose (possibly empty) boundary is made up of tori and Klein bottles, and X is a trivalent graph that is a spine of ∂M . A standard skeleton of a 3-manifold with marked boundary (M, X) is a standard sub-polyhedron P of M such that P ∩∂M coincides with X and with ∂P , and such that P ∪ ∂M is a spine of M \ B (where B is a ball). In this paper, we will prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate any two standard skeleta of a 3-manifold with marked boundary to each other. We will also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set of moves, made up of the MP-move and the L-move, suffices to relate any two standard skeleta of the 3-manifold with marked boundary to each other. For the 3-manifolds with marked boundary that do not fulfil this condition, we give two other moves: the CR-move and the T-move. The first one is local and, with the MP-move and the L-move, suffices to relate to each other any two standard skeleta of a 3-manifold with marked boundary fulfilling another condition. For the universal case, we will prove that the non-local T-move, with the MP-move and the L-move, suffices to relate to each other any two standard skeleta of a generic 3-manifold with marked boundary. As a corollary, we will get that disc-replacements suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary.