GENERALIZED GAUSSIAN SUMS CHERN-SIMONS-WITTEN-JONES INVARIANTS OF LEN-SPACES

Abstract

Starting from evaluating all Gaussian sums, we calculate {τ r , r≥2} (in the 4r-th cyclotomic fields) and {ξ r , r odd≥3} (in the r-th cyclotomic fields) for all lens spaces L(p, q). We prove that they are all algebraic integers and show that ξ r determines the Dedekind sum s(q, p), and hence determines the generalized Casson invariant of lens space. We conjecture these two properties hold for more general 3-manifold, and some evidences are discussed. Though the formulae are not simple, we are able to give necessary and sufficient conditions for lens spaces to have the same CSWJ invariants. Examples of lens spaces with the same invariants but different topological types are given. Some applications in number theory are also included.