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.
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