  The aim of this article is twofold: firstly, we show how to recover the
smooth Deligne-Beilinson cohomology groups from a Heegaard splitting of a
closed oriented smooth 3-manifold by extending the usual \tch-de Rham
construction; secondly, thanks to the above and still relying on a Heegaard
splitting, we explain how to compute the partition functions of the $U(1)$
Chern-Simons and BF theories.
