Simplekslerin çarpımları üzerindeki dar örtülerin sınıflandırılmaları

A small cover is a smooth closed manifold which admits a locally standard (Z/2)n -action whose orbit space is a simple convex polytope. Topological invariants of small covers can be obtained from the combinatorial structure of the quotient polytopes. Moreover, small covers are classified by their characteristic functions, which are enumerable. These properties make them favourable objects of topology. One of the important problems in this area is to classify small covers over given polytopes. Since the small covers over them corresponds to real Bott manifolds, the most intensively studied family of polytopes is cubes. A recursive formula for the number of (Z/2)n-equivariant equivalence classes of the small covers over cube is known. The aim of this project is to obtain a receursive formula for the number of weakly (Z/2)n-equivariant equivalence classes and the diffeomorphism classes of small covers over cubes. Note that the number of weakly (Z/2)n-equivariant equivalence classes gives an upper bound for the number of diffeomorphism classes. It is known that the diffeomophism classes of small covers over cubes corresponds to the equivalence classes of acyclic digraphs under some operations and these numbers are known up to n=8. Some of these operations on acyclic digraphs are firstly defined in these recent studies. It is also important to obtain such a formula in this sense. To find the number of the small cover over cubes, first, a database of small dimensions will be built by computations implemented with GAP. Then the results will be extended to the general case by analyzing the sample data and the obtained hypothesis will be proved combinatorically. By achieving goals of the project, we will obtain results which are most likely to be published in prestigious journals. The sociological-economical goal of this project is to supervise and to support a master student. Lastly, by organizing weekly seminars we are planning to introduce these subjects to other researchers in the region and to initiate new collaborations.