We investigate Tukey functions from the ideal of all closed nowhere dense subsets of 2N. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of N. We also prove non-existence of various special types of Tukey reductions from the nowhere dense ideal to analytic P-ideals. In connection with these res...

We show that the distance between a finite filling slope and a reducible filling slope on the boundary of a hyperbolic knot manifold is at most one. Let M be a knot manifold, i.e. a connected, compact, orientable 3-manifold whose boundary is a torus. A knot manifold is said to be hyperbolic if its interior admits a complete hyperbolic metric of finite volume. Let M(α) denote the manifold obtain...

IT HAS long been conjectured that surgery on a knot in S3 yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [7.8, 9, 10, 111). One may regard the Poenaru conjecture (solved in [S]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold A4 produces a reducible 3-manifold...

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≥ 0. We study J-P. Serre's notion of G-complete reducibility for subgroups of G. Specifically, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, gener...

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≥ 0. We study J-P. Serre's notion of G-complete reducibility for subgroups of G. Specifically, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, gener...

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≥ 0. We study J.-P. Serre's notion of G-complete reducibility for subgroups of G. In particular, for a subgroup H and a normal subgroup N of H, we look at the relationship between G-complete reducibility of N and of H, and show that these properties are equivalent if H/N is linearly reductive, gen...

Let M be a compact, orientable, irreducible, ∂-irreducible, anannular 3manifold with one component T of ∂M a torus. A slope r on T is a T isotopy class of essential, unoriented, simple closed curves on T , and the distance between two slopes r1 and r2, denoted by 4(r1, r2), is the minimal geometric intersection number among all the curves representing the slopes. For a slope r on T , we denote ...

A 3-manifold is ∂-reducible if ∂M is compressible in M . By definition, this means that there is a disk D properly embedded in M so that ∂D is an essential curve in ∂M . The disk D is called a compressing disk of ∂M , or a ∂-reducing disk of M . Now suppose M is a ∂-reducible manifold. Let K be a knot in a 3-manifold M such that ∂M is incompressible in M − K. A Dehn surgery on K is called ∂-red...