The disk complex \(\Gamma (H)\) of a surface \(H\) that is properly embedded, separating and with no torus components in a compact, orientable 3-manifold \(M\), is the complex whose vertices are isotopy classes of compression disks for \(H\) and an \(m\)-simplex is a set of \(m+1\) vertices whose representative compressions are pairwise disjoint. The disk complex was first defined by D. McCullough when \(H\) is the boundary of a handlebody, and he showed that in this case \(\Gamma (H)\) is contractible.
In this paper, the author defines \(H\) to be topologically minimal if \(\Gamma(H) \) is either empty or non-contractible, and then defines the topological index of \(H\) to be 0 if \(\Gamma (H)\) is empty and otherwise the smallest number \(n\) for which \(\pi_{n-1}(\Gamma(H))\) is non-trivial. Clearly \(H\) has topological index 0 if and only if it is incompressible. Now \(H\) has topological index 1 if and only if \(\pi_{0}(\Gamma(H))\) is non-trivial if and only if \(\Gamma (H)\) is disconnected. Using McCullough’s result, the author shows that \(H\) has topological index 1 if and only if it is strongly irreducible. The author then classifies the surfaces of topological index 2 as the surfaces (called critical surfaces) whose compressions can be partitioned into two sets in such a way that, in each set there are disjoint compression disks on opposite sides of \(H\) and whenever two compression disks from different sets are on opposite sides of \(H\) then they must intersect.
The main result of this paper asserts that a topological minimal surface \(H\) and an incompressible surface \(F\) in an irreducible 3-manifold \(M\) can be isotoped so that \(H - N(F)\) is topologically minimal in \(M - N(F)\). It follows that \(H\) and \(F\) can be isotoped so that every loop of \(H \cap F\) is essential in both \(H\) and \(F\).
As applications of topological index theory to Heegaard splittings of 3-manifolds, it is shown that if a Heegaard surface of a 3-manifold \(M\) is topological minimal, then \(\partial M\) is incompressible. As a corollary, if \(M\) contains a closed topological minimal surface \(H\), then either (1) \(M\) contains a non-boundary parallel, incompressible surface, (2) \(H\) is a Heegaard surface of \(M\), (3) \(H\) is contained in a ball, or (4) \(H\) is isotopic into a neighborhood of \(\partial M\).
In general the disjoint union of topological minimal surfaces may not be topological minimal, but the author shows that the converse is true: if the disjoint union is topologically minimal, then each component is topologically minimal; and the topological indices of the components sum up to the topological index of the union.
The paper concludes with a list of interesting questions and conjectures.