We prove a -set unknotting theorem for Nobeling spaces. This generalizes a result obtained by S. Ageev for a restricted class of -sets. The theorem is proved for a certain model of Nobeling spaces.
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On the dimension of Z-setsmath.GT
We offer a short and elementary proof that, for a Z-set A in a finite-dimensional ANR Y, dimA<dimY. This result is relevant to the study of group boundaries. The original proof by Bestvina and Mess relied on cohomological dimension theory.
We prove that Dranishnikov's -dimensional resolution is a UV-divider of Chigogidze's -dimensional resolution . This fact implies that preserves -sets. A further development of the concept of UV-dividers permits us to find sufficient conditions for $d_k^{-1}(…
Generalizes expansion and collapse theory to metric spaces.
problem Compactification of arbitrary metric spaces.
method Expands dual notions of expansion and collapse to arbitrary metric spaces and infinitely many moves.
result Proves compactification theorems, particularly for Z-set compactifications.
In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudo-collars, which are stackings of …
We define a new compactification of outer space (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a -set. The classical compactification made of very small -actions on -trees, however, fails to be locally -connected as soon as $N…
Extends geometric group theory techniques to arbitrary proper metric ARs.
problem Generalizing geometric group actions to non-freely acting groups with torsion.
method Extends techniques from geometric group theory to arbitrary proper metric ARs, eliminating freeness requirements.
result New theorems on proper homotopy equivalence and Z-structures for geometric actions.
A Z-structure on a group G, defined by M. Bestvina, is a pair (\hat{X}, Z) of spaces such that \hat{X} is a compact ER, Z is a Z-set in \hat{X}, G acts properly and cocompactly on X=\hat{X}\Z, and the collection of translates of any compact set in X forms a null sequence in \hat{X}. It is natural to ask whether a given…
The paper constructs coronas for combable spaces under specific conditions.
problem Constructing boundaries for combable spaces.
method Introducing properness, coherence, and expandingness for combings; constructing coronas using Rips complexes.
result Bijectivity of transgression maps, injectivity of the coarse assembly map, and surjectivity of the coarse co-assembly map for groups.
Metrizable uniform spacesmath.GT
Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…