No abstract

For any s ∈ [−∞, 0] and oriented homology 3-sphere Y , we introduce a homology cobordism invariant rs(Y ) whose value is in (0, ∞]. The values {rs(Y )} are contained in the critical values of the SU (2)-Chern-Simons functional of Y , and we have a negative definite cobordism inequality and a connected sum formula. As applications, we have several new results on the homology cobordism group. As one of such results, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. As another result, we show that if the 1-surgery of a knot has the Frøyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. In another direction, we use {rs} to define a filtration on the homology cobordism group which is parametrized by [0, ∞]. Moreover, as a hyperbolic example, we compute an approximate value of rs for the 1/2-surgery along the mirror of the knot 52 in Rolfsen's knot table. Z 6.2. A pseudometric on Ker h 1

The set RT(M) of values of the SL(2, C)-Reidemeister torsion of a 3-manifold M can be both finite and infinite. We prove that RT(M) is a finite set if M is the splice of two certain knots in the 3-sphere. The proof is based on an observation on the character varieties and A-polynomials of knots.

We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms in number theory play a key role. We also discuss the Alexander polynomial of homologically fibered knots.