In this note we show that sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces are homeomorphic to 3-manifolds. As a corollary we obtain the validity of the Borel conjecture for ...sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces.
An operation which assigns to an arbitrary family of sets the class of sets which can be translated away from every set from the fixed family is considered in abelian groups. Assuming CH it is proven ...that on the real line meager sets can be defined as sets “shiftable” from the family of strong measure zero sets (K=SMZ⁎). A similar result is shown for Lebesgue null sets and strongly meager sets (N=SM⁎).
Additionally a certain characterization of the family of meager-additive sets is given.
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in 6 holds while the classical Borel ...Conjecture fails.
There has recently been considerable interest in productively Lindelöf spaces, i.e. spaces such that their product with every Lindelöf space is Lindelöf. See e.g. 5,31,1,30,26,24,4, and work in ...progress by Brendle and Raghavan. Here we make several related remarks about such spaces. Indestructible Lindelöf spaces, i.e. spaces that remain Lindelöf in every countably closed forcing extension, were introduced in 28. Their connection with topological games and selection principles was explored in 27. We find further connections here.
An advanced treatment of surgery theory for graduate students and researchersSurgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. A Course on Surgery ...Theory offers a modern look at this important mathematical discipline and some of its applications. In this book, Stanley Chang and Shmuel Weinberger explain some of the triumphs of surgery theory during the past three decades, from both an algebraic and geometric point of view. They also provide an extensive treatment of basic ideas, main theorems, active applications, and recent literature. The authors methodically cover all aspects of surgery theory, connecting it to other relevant areas of mathematics, including geometry, homotopy theory, analysis, and algebra. Later chapters are self-contained, so readers can study them directly based on topic interest. Of significant use to high-dimensional topologists and researchers in noncommutative geometry and algebraic K-theory, A Course on Surgery Theory serves as an important resource for the mathematics community.
This book is the first to present a new area of mathematical
research that combines topology, geometry, and logic. Shmuel
Weinberger seeks to explain and illustrate the implications of the
general ...principle, first emphasized by Alex Nabutovsky, that
logical complexity engenders geometric complexity. He provides
applications to the problem of closed geodesics, the theory of
submanifolds, and the structure of the moduli space of isometry
classes of Riemannian metrics with curvature bounds on a given
manifold. Ultimately, geometric complexity of a moduli space forces
functions defined on that space to have many critical points, and
new results about the existence of extrema or equilibria follow.
The main sort of algorithmic problem that arises is recognition: is
the presented object equivalent to some standard one? If it is
difficult to determine whether the problem is solvable, then the
original object has doppelgängers--that is, other objects that are
extremely difficult to distinguish from it. Many new questions
emerge about the algorithmic nature of known geometric theorems,
about "dichotomy problems," and about the metric entropy of moduli
space. Weinberger studies them using tools from group theory,
computability, differential geometry, and topology, all of which he
explains before use. Since several examples are worked out, the
overarching principles are set in a clear relief that goes beyond
the details of any one problem.
This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general ...principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow. The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgängers--that is, other objects that are extremely difficult to distinguish from it. Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems, " and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.
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