# Is there any 1-cusp 3-manifold without Lens space Dehn filling?

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?

"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.