Ideal triangulations of 3‐manifolds up to decorated transit equivalences

Riccardo Benedetti

Mathematische Nachrichten, 2021, vol. 294, issue 2, 236-264

Abstract: We consider 3‐dimensional pseudo‐manifolds M̂ with a given set of marked point V such that M̂∖V is the interior of a compact 3‐manifold with boundary M. An ideal triangulation T of (M̂,V) has V as set of vertices. A branching (T,b) enhances T to a Δ‐complex. Branched triangulations of (M̂,V) are considered up to the b‐transit equivalence generated by isotopy and ideal branched moves which keep V pointwise fixed. We extend a well known connectivity result for “naked” ideal triangulations by showing that branched ideal triangulations of (M̂,V) are equivalent to each other. A pre‐branching (T,ω) is a system of transverse orientations at the 2‐facets of T verifying a certain global constraint; pre‐branchings are considered up to a natural pb‐transit equivalence. If M is oriented, every branching (T,b) induces a pre‐branching (T,ωb) and every b‐transit induces a pb‐transit. The quotient set of pre‐branchings up to transit equivalence is far to be trivial; we get some information about it and we characterize the pre‐branchings of the type ωb. Pre‐branched and branched moves are naturally organized in subfamilies which give rise to restricted transit equivalences. In the branching setting we revisit, with some complement, early results about the sliding transit equivalence and outline a (partially conjectural) conceptually different approach to the branched ideal connectivity and eventually also to the naked one. The basic idea is to point out some structures of differential topological nature on M which are carried by every branched ideal triangulation (T,b) of M̂, are preserved by the sliding transits and can be modified by the full branched transits. The non ambiguous transit equivalence already widely studied on pre‐branchings lifts to a specialization of the sliding equivalence on branched triangulations; we point out a few specific insights, again in terms of carried structures preserved by the non ambiguous and which can be modified by the whole sliding transits.

Date: 2021
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