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
Abstract

It is known that for coprime integers $p>q\ge 1$, the lens space $L\left({p}^{2},pq-1\right)$ bounds a rational ball, ${B}_{p,q}$, arising as the $2$-fold branched cover of a (smooth) surface in ${B}^{4}$ bounding the associated $2$-bridge knot or link. Lekili and Maydanskiy give handle decompositions for each ${B}_{p,q}$; whereas, Yamada gives an alternative definition of rational balls, ${A}_{m,n}$, bounding $L\left({p}^{2},pq-1\right)$ 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 ${A}_{m,n}$ admits a Stein filling of the universally tight contact structure, ${\stackrel{̄}{\xi }}_{st}$, on $L\left({p}^{2},pq-1\right)$ 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 ${B}_{p,q}$ and ${A}_{m,n}$.

Keywords
4-manifolds, handle calculus, rational blow-down
Primary: 57R65
Secondary: 57R17