A partial order on multibranched surfaces in 3-manifolds (Q2295667)

The author studies the union of a finite collection of compact surfaces with boundaries identified, either by homeomorphisms or covering maps. Such a 2-dimensional complex is called a multibranched surface. In such a complex, every point has a neighborhood homeomorphic either to \(E^2\) or to the union of a finite number of upper half planes with their boundaries identified. Such complexes are hypothesized to be embedded essentially in a 3-manifold, and suitably normalized (maximally spread). Under these hypotheses equivalence classes are shown to admit a partial ordering. In addition, a class with an atoroidal and acylindrical representative is shown to be minimal in this ordering. The paper ends with an illuminating example.