Cyclic Suspension of Knots and Periodicity of Signature for Singularities

By a knot we mean a pair (S, M~) with M~ a smooth closed oriented submanifold of S (m^3). If such a knot is given and i:S-> gm+z j s t j ^ standard embedding, then one can isotope / in an essentially unique way (Lemma 1 below) to an embedding j :s -+S whose intersection with iS is M c S transversally. The «-fold cyclic branched cover of (S, iS) branched along (jS, M~) exists uniquely and is a manifold pair (S%, MTM)9 where STM +2 is diffeomorphic to the sphere. This pair we call the n-fold cyclic suspension of (S, M~), or briefly nsuspension. This construction is motivated by the following theorem. Recall that if g:(C, 0)—•((?, 0) is a polynomial with isolated singularity at zero, the link Kg^S*' of g is the intersection of g"^) with a sufficiently small sphere S~^ C at the origin.