General Topology-I

. We give a new definition of the derived category of constructible Q`-sheaves on a scheme, which is as simple as the geometric intuition behind them. Moreover, we define a refined fundamental group of schemes, which is large enough to see all lisse Q`-sheaves, even on non-normal schemes. To accomplish these tasks, we define and study the pro-etale topology, which is a Grothendieck topology on schemes that is closely related to the etale topology, and yet better suited for infinite constructions typically encountered in `-adic cohomology. An essential foundational result is that this site is locally contractible in a well-defined sense.

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The Pro-étale topology for schemes

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

Virtual Moduli Cycles and Gromov-Witten Invariants of Noncompact Symplectic Manifolds

This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can favorably be played for locally compact groups. The development of the theory is illustrated by numerous examples, including matrix groups with entries in the the field of real or complex numbers, or other locally compact fields such as p-adic fields, isometry groups of various metric spaces, and, last but not least, discrete group themselves. Word metrics for compactly generated groups play a major role. In the particular case of finitely generated groups, they were introduced by Dehn around 1910 in connection with the Word Problem. Some of the results exposed concern general locally compact groups, such as criteria for the existence of compatible metrics on locally compact groups. Other results concern special classes of groups, for example those mapping onto the group of integers (the Bieri-Strebel splitting theorem for locally compact groups). 
Prior to their applications to groups, the basic notions of coarse and large-scale geometry are developed in the general framework of metric spaces. Coarse geometry is that part of geometry concerning properties of metric spaces that can be formulated in terms of large distances only. In particular coarse connectedness, coarse simple connectedness, metric coarse equivalences, and quasi-isometries of metric spaces are given special attention. The final chapters are devoted to the more restricted class of compactly presented groups, generalizing finitely presented groups to the locally compact setting. They can indeed be characterized as those compactly generated locally compact groups that are coarsely simply connected.

Metric Geometry of Locally Compact Groups

We review some selected recent results concerning selection principles in topology and their relations with several topological constructions.

Selected results on selection principles

[i]Topological Groups and Related Structures[/i] provides an extensive overview of techniques and results in the topological theory of topological groups. This overview goes sufficiently deep and is detailed enough to become a useful tool for both researchers and students. This book presents a large amount of material, both classic and recent (on occasion, unpublished) about the relations of Algebra and Topology. It therefore belongs to the area called Topological Algebra. More specifically, the objects of the study are subtle and sometimes unexpected phenomena that occur when the continuity meets and properly feeds an algebraic operation. Such a combination gives rise to many classic structures, including topological groups and semigroups, paratopological groups, etc. Special emphasis is given to tracing the influence of compactness and its generalizations on the properties of an algebraic operation, causing on occasion the automatic continuity of the operation. The main scope of the book, however, is outside of the locally compact structures, thus distinguishing the monograph from a series of more traditional textbooks. The book is unique in that it presents very important material, dispersed in hundreds of research articles, not covered by any monograph in existence. The reader is gently introduced to an amazing world at the interface of Algebra, Topology, and Set Theory. He/she will find that the way to the frontier of the knowledge is quite short — almost every section of the book contains several intriguing open problems whose solutions can contribute significantly to the area. – Contents : – Introduction to Topological Groups and Semigroups; – Right Topological and Semitopological Groups; – Topological Groups: Basic Constructions; – Some Special Classes of Topological Groups; – Cardinal Invariants of Topological Groups; – Moscow Topological Groups and Completions of Groups; – Free Topological Groups; – Factorizable Topological Groups; – Compactness and its Generalizations in Topological Groups; – Actions of Topological Groups on Topological Spaces.

Alexander Arhangel’skii

Papers

Topological Groups and Related Structures

Paratopological and semitopological groups versus topological groups

Relative topological properties and relative topological spaces

Remainders in compactifications and generalized metrizability properties

Remainders of rectifiable spaces