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By P. Hoffman, V. Snaith

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Since for each 3-residue T, br - vT/2 < 2, it follows that ET(br - vT/2) < 2tG. As the sum on the left is 2bG - 2vG, we get bG < VG + tG. ■ Corollary 1 Let G be a (3 + 1)-graph having bG 2-residues, to 3-residues and vG vertices. Then, VG + tG > bG. Proof: A direct consequence of the proof of the previous proposition. 2 A Basic Lemma Since the construction of the H* below is easily implementable , the next lemma provides a useful decision tool for testing whether a given identification scheme produces a 3-manifold.

This means that the identifications in (G, S2) induce a perfect matching of the vertices of H. In the above example: {lbX,2aY}, {lbY,2aX}, {laY,2bX}, {laX, 2bY}. Add new edges, of color 3 to H linking these mates, naming the resulting graph H. In the example at hand, H' is isomorphic, up to a switching of colors 0 and 3, to Fig. 26: The superattractor for RP3 We prove, in Lemma 1 of next section , that this general construction produces in this case a 3-gem inducing RP3. Since there are no other 3-gems with eight or less vertices inducing RP3, the above 3-gem is the superattractor for RP3.

Thus the vertices of G are of two types clockwise and anticlockwise and every edge has ends of distinct type. Thus G is bipartite. Conversely assume that G is bipartite. Orient the edges consistently from one class of the bipartition to the other (from black to white). 2. 3-Manifolds from 3-Gems 19 Fig. 21: Orientable embedding from the bipartition Embed the 01-gons so that the oriented 0-colored edges (0-darts) point clockwise; embed the 12-gons so that the 1-darts point clockwise; finally, embed the 20-gons so that the 2-darts point clockwise.

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