Title: Cosmetic Surgery Conjecture on S^3
Abstract: It has been known over 40 years that every closed orientable 3-manifold is obtained by surgery on a link in S^3. However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description. In this talk, we discuss the following uniqueness theorem for Dehn surgey on a nontrivial knot in S^3. Let K be a knot in S^3, and let r and r' be two distinct rational numbers of same sign, allowing r to be infinite; then there is no orientation preserving homeomophism between the manifolds obtained by performing Dehn surgery of type r and r', respectively. In particular, this result implies the famous Knot Complement Theorem of Gordon and Luecke.