A second cohomology class of the symplectomorphism group with the discrete topology

2018 ◽  
Vol 18 (1) ◽  
pp. 187-219
Author(s):  
Ryoji Kasagawa
Keyword(s):  
Cohomology Class ◽  
Discrete Topology ◽  
Symplectomorphism Group

scholarly journals Elements of finite order in Stone-Čech compactifications

1993 ◽  
Vol 36 (1) ◽  
pp. 49-54 ◽  
Author(s):  
John Baker ◽  
Neil Hindman ◽  
John Pym
Keyword(s):  
Compact Group ◽  
Finite Order ◽  
Free Semigroup ◽  
Continuous Homomorphism ◽  
Discrete Topology ◽  
Natural Extensions ◽  
Finite Sums

Let S be a free semigroup (on any set of generators). When S is given the discrete topology, its Stone-Čech compactification has a natural semigroup structure. We give two results about elements p of finite order in βS. The first is that any continuous homomorphism of βS into any compact group must send p to the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.

scholarly journals Mapping i2 on the free paratopological groups

2017 ◽  
Vol 101 (115) ◽  
pp. 213-221
Author(s):  
Fucai Lin ◽  
Chuan Liu
Keyword(s):  
Topological Space ◽  
Nonnegative Integer ◽  
Discrete Topology ◽  
Natural Mapping ◽  
Closed Mapping ◽  
Paratopological Group

Let FP(X) be the free paratopological group over a topological space X. For each nonnegative integer n ? N, denote by FPn(X) the subset of FP(X) consisting of all words of reduced length at most n, and in by the natural mapping from (X ? X?1 ? {e})n to FPn(X). We prove that the natural mapping i2:(X ? X?1 d ?{e})2 ? FP2(X) is a closed mapping if and only if every neighborhood U of the diagonal ?1 in Xd x X is a member of the finest quasi-uniformity on X, where X is a T1-space and Xd denotes X when equipped with the discrete topology in place of its given topology.

THE MINIMAL GENUS PROBLEM IN RULED MANIFOLDS

2008 ◽  
Vol 17 (04) ◽  
pp. 471-482
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Cohomology Class ◽  
Standard Form ◽  
Geometric Construction ◽  
Diffeomorphism Group ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H2(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H2(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

Symplectic (-2)-spheres and the symplectomorphism group

2021 ◽  
Author(s):  
Jun Li ◽  
Tianjun Li ◽  
Weiwei Wu
Keyword(s):  
Symplectomorphism Group

On Quasi Discrete Topological Spaces in Information Systems

2012 ◽  
Vol 3 (2) ◽  
pp. 38-52 ◽  
Author(s):  
Tutut Herawan
Keyword(s):  
Information System ◽  
Information Systems ◽  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Topological Spaces ◽  
Discrete Topology

This paper presents an alternative way for constructing a topological space in an information system. Rough set theory for reasoning about data in information systems is used to construct the topology. Using the concept of an indiscernibility relation in rough set theory, it is shown that the topology constructed is a quasi-discrete topology. Furthermore, the dependency of attributes is applied for defining finer topology and further characterizing the roughness property of a set. Meanwhile, the notions of base and sub-base of the topology are applied to find attributes reduction and degree of rough membership, respectively.

scholarly journals Block Extensions, Local Categories and Basic Morita Equivalences

2020 ◽  
Vol 71 (2) ◽  
pp. 703-728
Author(s):  
Tiberiu Coconeţ ◽  
Andrei Marcus ◽  
Constantin-Cosmin Todea
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cohomology Class ◽  
Group Extension ◽  
Modular System ◽  
Pointed Group ◽  
A Finite Group ◽  
Nilpotent Blocks ◽  
The Way

Abstract Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

scholarly journals Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1072 ◽  
Author(s):  
Sang-Eon Han ◽  
Saeid Jafari ◽  
Jeong Kang
Keyword(s):  
Applied Sciences ◽  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Discrete Topology ◽  
Separation Axiom ◽  
Different Types

The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n ∈ N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.

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