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If there is a pair of edges a, a such that the straight arc α connecting their midpoints is an essential, disconnecting curve in S, then we can take c0 to be α. In this case, it intersects each edge of H at most once. If none of the above cases holds, each edge is paired with a neighboring edge. In this case, pick four neighboring edges a, a , b, b , with a, a and b, b paired, and take c0 to be the union of the straight arcs connecting the midpoints of a and b and a and b . In this case, it intersects each edge of H at most twice.

Uk/gtp/. 19. M. Lackenby, ‘The volume of hyperbolic alternating link complements’ with an appendix by Ian Agol and Dylan Thurston, Proc. London Math. Soc. (3) 88 (2004) 204–224. 20. V. P. Leksin and A. V. Chernavskii, ‘Unrecognizability of manifolds. On Novikov’s theorem on the unrecognizability of the sphere S n for n 5’, Dokl. Math. 391 (2003) 89. 21. C. Lescop, Global surgery formula for the Casson–Walker invariant, Annals of Mathematics Studies 140 (Princeton University Press, Princeton, NJ, 1996).

C. Lescop, Global surgery formula for the Casson–Walker invariant, Annals of Mathematics Studies 140 (Princeton University Press, Princeton, NJ, 1996). 22. H. Levine, Classifying immersions into R4 over stable maps of 3-manifolds into R2 , Lecture Notes in Mathematics 1157 (Springer, Berlin, 1985). 23. H. Levine, ‘Stable mappings of 3-manifolds into the plane’, Singularities, Warsaw, 1985, Banach Center Publications 20 (PWN, Warsaw, 1988) 279–289. 24. A. Markov, ‘The insolubility of the problem of homeomorphy’, Dokl.

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3-manifolds efficiently bound 4-manifolds by Costantino F., Thurston D.


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