Romanskyi M. Functors and asymptotic properties of metric spaces.

In the thesis, the coarse and Lipschitz equivalence between some functorial constructions, as well as some properties of cone, join and suspension the in asymptotic categories are investigated. The thesis is devoted to the modern field of topology, which has been intensively developing in recent decades - asymptotic topology (the term “coarse geometry” is also used). It is devoted to the study of large-scale invariants of metric spaces and, more generally, coarse structures (modifications of the latter are the so-called balleans, which were introduced and studied by I.V. Protasov). This branch of topology also has its origins in the geometric group theory. Thus M. Gromov defined the concept of the asymptotic dimension of a finitely generated group, which has found application in solving certain open problems of algebraic topology. Modifications of the asymptotic Gromov dimension (asymptotic Assouad-Nagata dimension, asymptotic power dimension) were also identified. Naturally, the study of the asymptotic properties of metric spaces requires their inclusion in a certain category. The most important for application are the asymptotic categories of A. Dranishnikov and J. Roe, for which various functorial constructions have been identified and studied. One of the important tasks of asymptotic topology is the classification of functorial structures up to Lipschitz equivalence. The results of the thesis lie in this direction. We shown that the asymptotic cone CR+ and the suspension ∑R+ are not isomorphic. Dranishnikov defined the join X*Y as a subspace of the space of probability measures P2 (X∨Y) and raised the question of the isomorphism of the cone CX and the join X*R+ in the asymptotic category. We have proved that these spaces are not isomorphic, but we have established the isomorphism of the join X*R+ and the Cartesian product X×R+, for cases where X is the n-dimensional Euclidean space or a γ-slightly convex and δ-weakly concave geodesic space. From the results of the thesis it is worth noting those related to the coarse equivalence (i.e. equivalence in the asymptotic category of J. Roe) of functorial constructions. For example, the hyperspaces (spaces of compact subsets) of Euclidean spaces are considered and it is shown that they are not coarsely equivalent to hyperspaces of continua (connected compacta) and hyperspaces of convex compacta. The hyperspace exp2 R^m and the space R^m×Cone(R P^(m-1) ) are Lipschitz equivalent. This result can be considered a coarse analogue of one Schori's result. In the thesis it is also proved an asymptotic analogue of R. Bott's theorem.