The goal of this paper is to describe and clarify as much as possible the
3-dimensional topology underlying the Helmholtz cuts method, which occurs in a
wide theoretic and applied literature about Electromagnetism, Fluid dynamics
and Elasticity on domains of the ordinary space. We consider two classes of
bounded domains that satisfy mild boundary conditions and that become "simple"
after a finite number of disjoint cuts along properly embedded surfaces. For
the first class (Helmholtz), "simple" means that every curl-free smooth vector
field admits a potential.