  We consider the Laplace equation in the exterior of a thin filament in
$\mathbb{R}^3$ and perform a detailed decomposition of a notion of slender body
Neumann-to-Dirichlet (NtD) and Dirichlet-to-Neumann (DtN) maps along the
filament surface. The decomposition is motivated by a filament evolution
equation in Stokes flow for which the Laplace setting serves as an important
toy problem. Given a general curved, closed filament with constant radius
$\epsilon>0$, we show that both the slender body DtN and NtD maps may be
decomposed into the corresponding operator about a straight, periodic filament
plus lower order remainders. For the straight filament, both the slender body
NtD and DtN maps are given by explicit Fourier multipliers and it is
straightforward to compute their mapping properties. The remainder terms are
lower order in the sense that they are small with respect to $\epsilon$ or
smoother. While the strategy here is meant to serve as a blueprint for the
Stokes setting, the Laplace problem may be of independent interest.
