By Costantino F., Thurston D.

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If χ(S ) = 0 or if S is orientable and χ(S ) = −1, the proof is complete. 7 below to S . Note that χ(S ) = χ(S) − n. 7. Let S be a connected surface with boundary, with χ(S) < 0 and not a thricepunctured sphere, equipped with a set H of essential, pairwise nonparallel arcs. Then it is possible to apply O(−χ(S)) 0 → 2-moves to H and ﬁnd a set of curves in S intersecting H at most −18χ(S) times and cutting S into annuli of type 2 in the above list and two surfaces S1 and S2 such that |H1 (Si ; Z2 )| < |H1 (S; Z2 )| for i = 1, 2.

Zilber, ‘Semi-simplicial complexes and singular homology’, Ann. of Math. (2) 51 (1950) 499–513. ´ 7. M. Gromov, ‘Volume and bounded cohomology’, Inst. Hautes Etudes Sci. Publ. Math. 56 (1982) 5–99. 8. M. Gromov, ‘Singularities, expanders and topology of maps. part 1: Homology versus volume in the spaces ´ of cycles’, Preprint, Institut des Hautes Etudes Scientiﬁques, 2007. 9. J. Hass and J. C. Lagarias, ‘The minimal number of triangles needed to span a polygon embedded in Rd ’, Discrete and computational geometry: The Goodman–Pollack festschrift, Algorithms and Combinatorics 25 (Springer, Berlin, 2003) 509–526.

16. S. King, ‘Polytopality of triangulations’, PhD Thesis, Universit´e Louis Pasteur, Strasbourg, June 2001. 17. R. Kirby and P. Melvin, ‘Evaluations of the 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2, C)’, Geometry of low-dimensional manifolds, 2, Durham, 1990 (ed. S. K. Donaldson and C. B. Thomas), London Mathematical Society Lecture Notes Series 151 (Cambridge University Press, Cambridge, 1990) 101–114. 18. R. Kirby and P. Melvin, ‘Local surgery formulas for quantum invariants and the Arf invariant’, Proceedings of the Casson Fest (ed.

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