Vol. 289, No. 1, 2017

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On handlebody structures of rational balls

Luke Williams

Vol. 289 (2017), No. 1, 203–234
DOI: 10.2140/pjm.2017.289.203

It is known that for coprime integers p > q 1, the lens space L(p2 ,pq 1) bounds a rational ball, Bp,q, arising as the 2-fold branched cover of a (smooth) surface in B4 bounding the associated 2-bridge knot or link. Lekili and Maydanskiy give handle decompositions for each Bp,q; whereas, Yamada gives an alternative definition of rational balls, Am,n, bounding L(p2 ,pq 1) by their handlebody decompositions alone. We show that these two families coincide, answering a question of Kadokami and Yamada. To that end, we show that each Am,n admits a Stein filling of the universally tight contact structure, ξ̄st, on L(p2 ,pq 1) investigated by Lisca. Furthermore, we construct boundary diffeomorphisms between these families. Using the carving process, pioneered by Akbulut, we show that these boundary maps can be extended to diffeomorphisms between the spaces Bp,q and Am,n.

4-manifolds, handle calculus, rational blow-down
Mathematical Subject Classification 2010
Primary: 57R65
Secondary: 57R17
Received: 26 April 2016
Revised: 23 December 2016
Accepted: 25 December 2016
Published: 12 May 2017
Luke Williams
Department of Mathematics
Kansas State University
Manhattan, KS 66502
United States