# Ideal triangulations of 3-manifolds up to decorated transit equivalences

## TL;DR

This paper proves that all branched ideal triangulations of a 3-manifold with marked points are equivalent under certain moves, extending known connectivity results and exploring structures preserved by these moves.

## Contribution

It extends connectivity results for branched ideal triangulations and introduces a new perspective based on differential topological structures preserved by transit moves.

## Key findings

- Branched ideal triangulations are all equivalent under b-transit moves.
- Pre-branchings are classified up to pb-transit equivalence.
- Structures of differential topology are preserved by certain transit moves.

## Abstract

We consider 3-dimensional pseudo-manifolds M with a given set of marked point V such that M-V is the interior of a compact 3-manifold with boundary. An ideal triangulation T of (M, V ) has V as its set of vertices. A branching (T, b) enhances T to a Delta-complex. Branched triangulations of (M, 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 triangulations by showing that branched ideal triangulations of (M, V) are equivalent to each other. A pre-branching 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 b induces a pre-branching w(b) 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 type w(b). Pre-branched and branched moves are naturally organized in subfamilies which give rise to restricted transit equivalences. In the branching setting we revisit early results about the sliding transit equivalence and outline a conceptually different approach to the branched connectivity and eventually also to the naked one. The basic idea is to point out some structures of differential topological nature which are carried by every branched ideal triangulation, are preserved by the sliding transits and can be modified by the whole branched transits. The non ambiguous transit equivalence already widely studied on pre-branchings lifts to a specialization of the sliding equivalence on branchings; 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.