In recent years quite a few attempts have been made to bring Lie theory to the computer, e.g. see van Leeuwen, Cohen and Lisser [.Cohen LiE.], Moody, Patera and Rand [.simpLie.], Bödi and Joswig [.boedi Joswig.], not to mention the various packages devoted to the study of finite groups of Lie type. When it comes to solving problems arising from real Lie groups by means of a computer frequently the derived linearized problems are studied instead. This means one tries to translate the initial problem about Lie groups into a (usually simpler one) about Lie algebras. However, the correspondence between Lie groups and Lie algebras is not 1--1. In fact, for a given Lie algebra g there are usually several (not isomorphic) groups having g as their associated Lie algebra. They are only locally isomorphic, which means that they are algebraically somehow similar but differ with respect to their global topological properties. Roughly speaking, during the linearization process the global topological information is lost.
Here we want to suggest an approach with is in some sense complementary. Instead of skipping the topological information we intend to focus on it.
We want to look for a computational method to solve problems of the following kind and related ones: Given a Lie group Gamma and a homogeneous manifold M. Does there exist a (continuous) transitive action of Gamma on M?
Without any doubt this immediately raises at least two questions. How should a non-discrete Lie group (being of uncountable cardinality as a set) be represented in the computer? How can one even expect that a question like this is decidable at all? Instead of trying to solve the problem in this generality we want to discuss a certain approximation of the problem which can be outlined as follows. A transitive action of Gamma on M gives rise to an infinitely long exact sequence of their respective homotopy groups (see below). The group Gamma and the manifold M are described by some of their algebraic and topological properties such as the dimension, information about compactness, maybe some homotopy groups which are known etc. It is explicitly admitted that only partial information will be given. Assume that a transitive action exists. Select finitely many among the infinitely many pieces of information obtained from the long exact homotopy sequence. Translate this into an arithmetical problem. Check for a contradiction. In case we actually arrive at a contradiction then this falsifies the assumption concerning the existence of a transitive action. However, if we do not obtain a contradiction, we might not gain any information.
