  This paper studies U(1)-Chern-Simons theory and its relation to a
construction of Chris Beasley and Edward Witten. The natural geometric setup
here is that of a three-manifold with a Seifert structure. Based on a
suggestion of Edward Witten we are led to study the stationary phase
approximation of the path integral for U(1)-Chern-Simons theory after one of
the three components of the gauge field is decoupled. This gives an alternative
formulation of the partition function for U(1)-Chern-Simons theory that is
conjecturally equivalent to the usual U(1)-Chern-Simons theory. The goal of
this paper is to establish this conjectural equivalence rigorously through
appropriate regularization techniques. This approach leads to some rather
surprising results and opens the door to studying hypoelliptic operators and
their associated eta invariants in a new light.
