By Costantino F., Thurston D.
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Additional info for 3-manifolds efficiently bound 4-manifolds
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.
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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.
3-manifolds efficiently bound 4-manifolds by Costantino F., Thurston D.