A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems that arise naturally in fields such as computational geometry, graphics, robotics, social science, structural biology, and chemistry, using methods from computable topology.
[1][2][3] A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems.
It is known that the full classification of 3-manifolds can be done algorithmically,[10] in fact, it is known that deciding whether two closed, oriented 3-manifolds given by triangulations (simplicial complexes) are equivalent (homeomorphic) is elementary recursive.
[15] Computation of homology groups of cell complexes reduces to bringing the boundary matrices into Smith normal form.
Although this is a completely solved problem algorithmically, there are various technical obstacles to efficient computation for large complexes.