Ideal triangulations of 3‐manifolds up to decorated transit equivalences

Abstract

We consider 3‐dimensional pseudo‐manifolds trueM̂ 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 trueM̂, 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.

Ideal triangulations of 3-manifolds up to decorated transit equivalences

“…The non ambiguos relation lifts to a specialization of the sliding one on branched triangulations; we point out a few specific insights into the branching theory, again in terms of carried structures preserved by the non ambiguous and which can be modified by the whole slidings transits. In [7] we have developed a 2D counterpart of the present paper. This theme also emerged in [5], Section 5 within a so called "holographic" approach to 3D non ambiguous structures.…”

“…Fix any 2D branching (K, b). By a main result of [7], there is a sequence of b-flips (K, b) ⇒ (ψ(K), ψ(b)). By using the corresponding naked sequence we get an instance of layered taut (T, ω) as above; by using the b-sequence, we get a layered branched triangulation (T, b).…”

“…In the bulk we see two typical leaves of H ∆ which are transverse to ∂M along A. The trace of H ∆ on each hexagonal face of ∆ is a tile of the 2D analogous of H ∆ (see [7]). This is illustrated on the bottom of Figure 21.…”

“…A main invariant of 3D non ambiguous classes of pre-branchings is the sliding equivalence class of (∂ T , ∂ω), hence ultimately the associated pair of oriented vertical and horizontal transvese foliations (with isolated singularities) on ∂ T . In fact we are considering here the branched triangulations of surfaces all together, with arbitrary number of vertices, under both sliding diagonal exchanges and 1 → 3 stellar moves (see [5] and [7]). If (T, b) is branched, we can apply the above constructions to the induced pre-branching ω b .…”

On ideal triangulations of surfaces up to branched transit equivalences

“…On the non ambiguous transit. In 3D the notion of non ambiguous structure, defined indeed in terms of transit of pre-branchings rather than of branchings [1], [4], gives rise to non trivial examples of intrinsic interest. In 2D the intrinsic content of the na-relation is not so evident.…”