Moves for standard skeleta of 3-manifolds with marked boundary

Mathematics – Geometric Topology

Scientific paper

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23 pages, 31 figures; Proposition 16 was false, so a non-orientable version of the T-move is added to correct the mistake, the

Scientific paper

We 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 to each other any two standard skeleta of a 3-manifold with marked boundary. We 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 to each other any two standard skeleta of the 3-manifold with marked boundary. For the 3-manifolds with marked boundary that do not fulfil this condition, we give three other moves: the CR-move, the T1-move and the T2-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 prove that the non-local T1-move and T2-move, with the MP-move and the L-move, suffice to relate to each other any two standard skeleta of a generic 3-manifold with marked boundary. As a corollary, we get that disc-replacements suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary.

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