The long-term objective of the project is to develop a new combined approach to complexity theory and to its applications in the algorithmic and computer topology of manifolds, including enumeration, classification, and computer recognition of 3-manifolds. The project also aims to establish new links with combinatorial group theory.

Three-manifolds are models of objects of the real world, and the theory of complexity already developed for them is natural and widely applicable. The plan is to found complexity theories for other types of geometric and algebraic objects and, most importantly, to investigate the interactions among these theories. This includes finding canonical triangulations of hyperbolic manifolds and orbifolds and developing a relative version of complexity theory. Another activity consists in elaborating algorithmic aspects of the theory of 2-complexes and investigating the Andrews-Curtis and Whitehead conjectures.

A crucial plan to be carried out within the project is the enumeration and recognition of 3-manifolds. This plan will be achieved with the aid of computers and will be concentrated on the class of manifolds with toral (possibly empty) boundary, which contains all closed Seifert and hyperbolic manifolds. Such a plan calls for a unified effort of specialists from several different branches of mathematics, namely low-dimensional topology, differential and hyperbolic geometry, and combinatorial group theory. This need is fully met by the substantial diversity in background and expertise of the members of the participating teams.

The most important foreseeable results are:

a) To achieve a clear understanding of the relationships between geometric and topological complexity. In particular, to find two-sided estimates of each one in terms of the other.
b) To prove the relative version of Gadgil's theorem on the relations between the Andrews-Curtis conjecture and algorithmic problems in combinatorial group theory.

c) To obtain a table of all closed 3-manifolds of complexity at least 10, which is a critical level of manifold enumeration.

d) To construct geometric and algebraic invariants sufficient to distinguish all the manifolds appearing in the tables.

e) To implement a partial recognition program for 3-manifolds which is effective up to complexity at least 50.

As customary in the mathematical community, it is planned to freely circulate the results and the information discovered. The tables and software mentioned in (c) and (e) will be made publicly available through the world wide web.