# On ideal triangulations of surfaces up to branched transit equivalences

## TL;DR

This paper studies the connectivity and equivalence classes of branched triangulations of closed surfaces, extending known results and introducing new invariants related to differential topology and foliations.

## Contribution

It extends connectivity results for triangulations to branched cases, characterizes equivalences under b-flips, and introduces invariants related to foliations and differential structures.

## Key findings

- Branched triangulations are equivalent if the surface has negative Euler characteristic and even c(S).
- Inversion of ambiguous edges connects different branchings under mild assumptions.
- Restricted b-flip subfamilies have complex quotient structures linked to topological invariants.

## Abstract

We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V point-wise fixed. We extend a well known connectivity result for `naked' triangulations; in particular in the generic case when S has negative Euler-Poincare' characteristic c(S), we show that branched triangulations are equivalent to each other if c(S) is even, while this holds also for odd c(S) possibly after the complete inversion of one of the two branchings. Moreover we show that under a mild assumption, two branchings on a same triangulation are connected via a sequence of inversions of ambiguous edges (and possibly the total inversion of one of them). A natural organization of the b-flips in subfamilies gives rise to restricted transit equivalences with non trivial (even infinite) quotient sets. We analyze them in terms of certain preserved structures of differential topological nature carried by any branched triangulations; in particular a pair of transverse foliations with determined singular sets contained in V, including as particular cases the configuration of the vertical and horizontal foliations of the square of an Abelian differential on a Riemann surface.