Math. Proc. Camb. Phil. Soc. (1994), 115, 229 229 Printed in Great Britain Topological methods to compute Chern-Simons invariants BY DAVID R. AUCKLY University of Texas, Austin, TX 78712-1082 USA (Received 21 September 1992; revised 3 June 1993) 1. Introduction In this paper we develop a method which may be used to compute the Chern—Simons invariants of a large class of representations on a large class of manifolds. This class includes all representations on all Seifert fiber spaces, all graph manifolds, and some hyperbolic manifolds. I owe many thanks to Peter Scott, John Harer, Frank Raymond, Ron Fintushel, Paul Kirk and Eric Klassen, without whose help and support this paper could not have been written. After covering some background material, we will compute the Chern—Simons invariants of all Seifert fiber spaces. The techniques we use are very natural for Seifert fiber spaces, but work equally well for many other manifolds after the techniques are translated into the language of surgery. We finish by translating the techniques into a sequence of moves which may be used to compute Chern—Simons invariants in a way similar to that in which skein relations may be used to compute knot invariants. First, we will define the Chern-Simons invariant and list some related facts and definitions. Throughout the paper we will use the group of unit quaternions, denoted by Sp r Geometrically, Spj is a 3-sphere with + 1 at the north pole and — 1 at the south pole. Two elements of Spj commute if and only if they lie on a great circle running through +1. The conjugacy classes are 2-spheres perpendicular to this family of great circles, and every element can be expressed as the commutator of two elements. The Chern-Simons invariant is defined for connections on the trivial Sp x bundle over a 3-manifold. Let w be a connection on the trivial Spj bundle over a closed 3- manifold. Definition. The Chern-Simons invariant of w is cs((o) =-j—£ cr*Re(w A rfw + |w A w A w), *± n JM where a is a section of the trivial bundle. Fact. The Chern-Simons invariant only changes by an integer if the section is changed or if w is replaced by a connection that differs by a gauge transformation. Fact. Flat connections on M/Gauge equivalence = Hom(7T 1 (Jf), Sp^/AdSpx- By the Chern-Simons invariant of a representation, we will mean the Chern-Simons invariant of the corresponding flat connection. Definition. A representation a: G-±Sp x is called reducible if its image is contained in a great circle.