By Francesco Costantino

We determine a calculus for branched spines of 3-manifolds via branched Matveev-Piergallini strikes and branched bubble-moves. We in brief point out a few of its attainable purposes within the research and definition of State-Sum Quantum Invariants.

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**Example text**

If |[x]E ∩ Bn |x < ∞, then {y ∈ [x]E ∩ Bn : D(y, x) ≥ D(y , x), ∀y ∈ [x]E ∩ Bn } is ﬁnite, non-empty and independent of x in its E-class (since D is a cocycle). It follows that there is a smooth invariant Borel set X0 ⊆ X such that for x ∈ X1 = X \ X0 and each n, |Bn ∩ [x]E |x = ∞. For each α ∈ NN we will deﬁne next an increasing sequence of fsr’s {Fnα }n∈N of E|X1 , so that B0 ∩X1 is a transversal for each Fnα and for each b ∈ B0 ∩X1 , [b]Fnα ⊆ B0 ∪ · · · ∪ Bn . We start with F0α = equality on B0 , α .

We then say that G generates E. It will be also convenient to consider another concept of graph, which for distinction we will call an L-graph (L stands for Levitt). This is simply a countable family Φ = {ϕi }i∈I of partial Borel isomorphisms, ϕi : Ai → Bi , where Ai , Bi are Borel subsets of X. We call Φ ﬁnite if I is ﬁnite.

An fsr (ﬁnite partial subequivalence relation) of a countable Borel equivalence relation E is a ﬁnite Borel equivalence relation F , deﬁned on a Borel set dom(F ) ⊆ X, such that F ⊆ E. Let [X]<∞ denote the standard Borel space of ﬁnite subsets of X, and let [E]<∞ denote the Borel subset of [X]<∞ of pairwise E-related ﬁnite subsets of X. Given a set Φ ⊆ [E]<∞ , we say that an fsr F ⊆ E is Φ-maximal if 1. ∀x ∈ dom(F )([x]F ∈ Φ) and 2. ∀S ∈ [X \ dom(F )]<∞ (S ∈ Φ). 3. Suppose E is a countable Borel equivalence relation and Φ ⊆ [E]<∞ is Borel.