The problem is undecidable for complexes of dimension 5 or more.
Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.
Thus, an ASC provides a finite representation of a geometric object.
Given an ASC, one can ask several questions regarding the topology of the GSC it represents.
The homeomorphism problem is: given two finite simplicial complexes representing smooth manifolds, decide if they are homeomorphic.