  In this article we use the combinatorial and geometric structure of manifolds
with embedded cylinders in order to develop an adiabatic decomposition of the
Hodge cohomology of these manifolds. We will on the one hand describe the
adiabatic behaviour of spaces of harmonic forms by means of a certain
\v{C}ech-de Rham complex and on the other hand generalise the
Cappell-Lee-Miller splicing map to the case of a finite number of edges, thus
combining the topological and the analytic viewpoint. In parts, this work is a
generalisation of works of Cappell, Lee and Miller in which a single-edged
graph is considered, but it is more specific since only the Gauss-Bonnet
operator is studied.
