This paper presents an approach to spatial representation and reasoning with lattices of topological relations. This approach is based on the Galois lattice theory that is used to create concepts, by associating sets of objects to sets of attributes. The objects considered here are the base relations of the region connection calculus RCC8 and the attributes are computational conditions that are used in geographic information systems for modeling topological relations. Three lattices, named TgE, TgL, and TgM, involving three sets of computational conditions, are introduced. The design and the properties of these lattices are detailed and then compared on the basis of the underlying computational conditions. Finally the three lattices are compared to each others, and to a fourth lattice, Tp, that is based on the set of all disjunctions of RCC8 relations, with respect to qualitative spatial reasoning capabilities.