By Alessandrini L.

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**Example text**

Then we have a natural bijection: CI(Q') \ {el, e2, e3} ---}CI(Q) \ {eo}. Moreover for ~/E 9V2(Q') we have that e E CI(Q') \ {el, e2, e3} is bad for 7 if and only if the corresponding edge of Q is bad for r The idea of the proof is to blow up bad edges and choose suitable liftings of the given fundamental 2-chain so that the total number of bad edges decreases. This idea must however be refined in particular to deal with bad loops. 4. 16: Proof of fact 2 continued We prove various facts which will lead to the conclusion.

I f P is a standard spine of a punctured Z-homology sphere then for every 5 E ~'I(P) there exists ~ E Jz2(P) with O~ = 5. 7 that every spine of a punctured Zhomology sphere admits oriented branchings. 7. Every oriented three-dimensional manifold with boundary admits a standard spine with an oriented branching. This result by itself is not very difficult and was already remarked in [21], even if not strictly in our context (Gillman and Rolfsen deal with closed manifolds only, and they accept simple spines).

Recall that 7r n OM is a trivalent graph in OM. 6. Branchings on P correspond in a natural and bijective way to subgraphs A of 7r-I(S(P)) n OM such that: (i) A consists of a disjoint union of circles; (ii) A contains exactly one of the three preimages of each open edge of S(P). 6. We confine ourselves to a sketch, leaving the details to the reader. By definition a branching is the choice of a preferred germ of disc along each edge, with the following compatibility condition: at each vertex one of the germs of disc is preferred for both its edges, and the opposite germ of disc is non-preferred for both its edges.