The lifting and classification problems for subspaces of covering spaces

While the fundamental group of a topological space is sufficient for the study of covering spaces, an algebraic invariant which encodes analogous information for subspaces of coverings has been lacking. Such subspaces are important in topology for various reasons: sometimes, one wants a compact subspace carrying the fundamental group of the covering space associated to a finitely generated (or presented) subgroup; other times, one wants to consider a fundamental domain for an action by deck transformations. We introduce the fundamental inverse monoid of a pointed topological space; the lifting and classification problems for subspaces of coverings are then solved using this tool.