The thesis for obtaining the Candidate of Physical and Mathematical Science (PhD) degree on the speciality 01.01.04 – geometry and topology. – Ivan Franko National University of Lviv, Lviv, 2021.
The thesis is devoted to the research of classes of infinite-dimensional manifolds, the model spaces of which are direct (injective) limits of absolute extensors in the categories of topological spaces. Haysey, Torunchyk, Sakai, Pentsak, Banakh and other mathematicians studied this type of manifolds.
We introduce a model space that is a direct limit of Tikhonov cubes, prove a characterization theorem for such a space, and establish some of its topological properties, in particular, topological homogeneity and local self-semilarity. These properties make possible to consider manifolds modeled on the direct limits of Tikhonov cubes. In the thesis, we prove a characterization theorem for such manifolds, as well as theorems on open and closed embedding in the model space.
Also, a strongly countable-dimensional analogue of the direct limit of Tikhonov cubes is considered, and characterization theorems are also proved for the manifolds modeled over this analogue. Finally, a strongly countable-dimensional resolvent of injective-Tikhonov manifoldsnis is contstructed.
The preservation by functorial constructions of infinite-dimensional manifolds which are locally strongly universal spaces for the class of matrizable compact spaces with a finite finite-dimensional derivative, has been studied (the theory of such manifolds was elaborated by T. Banakh). Here we consider the functors in the category of Tychonov spaces that are extensions of some normal (in the sense of E.V. Shchepin) functors of finite degree in the category of compact Haussdorff spaces. The functors of symmetric and hypersymmetric powers are examples of such functors.
Part of the results of the dissertation is dovoted to analogues in the category of -spaces of absorbing sets constructed by T. Radul for the class of compact metric spaces for which the transfinite expansion of the cover (Lebesgue) dimension does not exceed a given countable ordinal number. Characterization theorems and theorems on open embedding in the model space for such manifolds are proved.
A general construction of model spaces of infinite-dimensional topology is proposed, which allows to describe some classes of infinite-dimensional manifolds from a unique point of view and to unify some general theorems concerning such manifolds.
Keywords: injective (direct) limit, infinite-dimensional manifold, absolute extensor, universal mapping, characterization theorem, open embedding, Tikhonov cube, functor of finite degree, property C.
