Foundation of Shape Theory

The main goal of the project is to lay foundations and develop the intrinsic definition of shape. The theory of shape was founded about 40 years ago. Since that time more than a thousand research papers and several monographs ([1] ,[2], [3], [4]) on shape theory were published all over the world. The two classical approaches to shape theory are: 1) the original Borsuk's approach, based on embedding compact metric spaces in the Hilbert cube and studying the fundamental sequences of maps ([1], [2]) and 2) the inverse system approach ([3] and [4]). Both approaches use external spaces - neighbourhoods of spaces embedded in the Hilbert cube, ANR spaces and polyhedra. The intrinsic definition of shape does not use external spaces. The main tool in founding this third approach to shape theory are functions f : X →Y which are near to continuous functions, i.e., functions which are continuous up to some coverings of the space Y . The intrinsic definition of shape is defined using sequences or nets of such functions and cofinal families of coverings, i.e., families such that for any covering of the space Y, in the family there exists a finer element. Introducing the corresponding notion of homotopy, the shape morphisms are homotopy classes of sequences or nets of near continuous functions. The intrinsic definition of strong shape is introduced using homotopies of higher order [5]. Using this new definition of strong shape we expect to prove some new theorems in the theory of strong shape and solve some of the existing problems. Moreover, this new approach yields new proofs of already existing theorems of the strong shape theory. One of the main tasks of the proposed investigations will be to prove equivalence of the new definition and the already existing definition of strong shape. Use of the intrinsic definition of proper shape should be applicable to the shape of noncompact spaces, in particular, to locally compact metric spaces. Moreover, some applications to topological dynamics are expected. [1] K. Borsuk, THEORY OF SHAPE, Lecture Notes Series 28, Aarhus, 1971. [2] K.Borsuk, THEORY OF SHAPE, Polish Scientific Publishers, 1975. [3] S.Mardešić and J. Segal, SHAPE THEORY (The Inverse Sistem Approach), North-Holland Publishing Company, 1982 [4] S.Mardešić, STRONG SHAPE AND HOMOLOGY, Springer, 2000. [5] N. Shekutkovski, Intrinsic definition of strong shape for compact metrics spaces, Topology Proceedings, 39, 27-3