For a knot complement on 3-sphere, there's at least one Lens-space Dehn filling (1/0-slope). Is it also true for any 1-cusp 3-manifold?
1 Answer
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This question is not really research-level, but I suppose it might be hard to find explicitly in the literature.
"Most" 1-cusped hyperbolic 3-manifolds will not admit a lens space space filling (in an imprecise sense). The simplest obstruction is homology: if $H_1(M,\partial M;\mathbb{Q})\neq 0$ ($M$ has "cuspidal cohomology"), then any Dehn filing will have $H_1(M;\mathbb{Q})\neq 0$, so cannot be a lens space (if you regard $S^2\times S^1$ a lens space, then let $\dim H_1(M,\partial M;\mathbb{Q})>1$). There are infinitely many hyperbolic 1-cusped 3-manifolds $M$ with $H_1(M,\partial M;\mathbb{Q})$ any dimension that you like. I'll just refer to the Snappea census for a source of examples. Having $\dim H_1(M,\partial M;\mathbb{F}_p)\geq 2$ suffices as well.
Another obstruction is geometric. If in a maximal embedded horocusp, the length of all boundary components is $\geq 6$, then there is no Dehn filling with finite fundamental group, by a result of myself and Lackenby (actually, the longitude does not have to have length $\geq 6$, since Dehn filling will have $b_1 >0$). Take any hyperbolic knot, and take a $\geq 6$-fold cyclic cover to get examples.
Another obstruction is manifolds whose fundamental group has weight $\geq 3$. Then Dehn filling adds at most one relator to the group, so will have weight $\geq 2$, and hence will not have cyclic fundamental group (cyclic groups are weight 1). Of course, the homological obstructions given above give lower bounds on the weight, but I suspect that there are 1-cusped hyperbolic 3-manifolds which are homology circles and have arbitrarily large weight fundamental group.
One ought to be able to find other obstructions coming from just about any strong enough invariant: Alexander polynomial, TQFTs, Floer homology.
The ultimate form of this question is to give a complete characterization of hyperbolic knots in lens spaces. I think that all that one can say is that there is an algorithm (in principle) which will list all such manifolds. In fact, there's an algorithm that will take any manifold and determine if there's a Lens space Dehn filling. I'm not sure if this has been written down somewhere, but I could check if you like (or describe it in outline).
