Sums of lens spaces bounding rational balls

Abstract

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot.57M99; 57M25

Linear Independence in the Rational Homology Cobordism Group

“…From Theorem 1.1 it follows immediately that Θ 3 Q /Θ 3 Z contains subgroups isomorphic to Z ∞ , a fact first shown in [KL14] following work in [HLR12]. Another proof appears in [AL18] using the description of the subgroup of Θ 3 Q generated by lens spaces from [Lis07].…”

“…These groups have been extensively studied (e.g. [CH81,Frø96,OSz03,Lis07]) and are still a remarkable source of open problems (e.g. Problem 4.5 in [Kir97]); for instance, it is not yet known if there are n-torsion elements in Θ 3 Q for n = 2, or if there are Q-summands or quotients in Θ 3 Q .…”

The concordance genus of a knot, II

“…Many new methods have been applied to obstruct torsion in C , and these have resolved many of the basic examples taken from the table of low-crossing number knots. See, for example, Grigsby, Ruberman and Strle [9], Jabuka and Naik [11], Lisca [14] and Livingston and Naik [17]. New test cases can be found by examining such algebraic concordance relations.…”

On Seifert fibered spaces bounding definite manifolds

“…Indeed, in this context Donald Much like the proofs by Donald and Aceto, our proof also proceeds by means of Donaldson's theorem. However, their proofs rely on the work of Lisca [Lis07] which gives a detailed analysis on sums of linear lattices embedding in a full-rank lattice. We give a short proof of Theorem 4 circumventing the reliance on Lisca's work.…”