局部紧群的Gap研究和群拓扑的构造

局部紧拓扑群的研究始于19世纪Sophus Lie的工作，局部紧群与数论、分析、表示论等领域有密切的关系。任意一个局部紧群可以看作它的连通分支和相应的商群（完全不连通局部紧群）的扩张。另一方面，拓扑群与序结构也有密切的关系。一个抽象群上的拓扑群拓扑按照包含序构成一个完备格。因此从序结构的角度研究拓扑群的拓扑性质和代数性质成为拓扑群理论研究的新热点。本项目拟研究局部紧群G上拓扑τ在G的群拓扑格中的Gap问题，具体研究内容如下：. 1.研究连通局部紧群和完全不连通局部紧群的Gap问题，即讨论该群上的拓扑在相应的群拓扑格中的前趋和后继的存在与构造问题；. 2.研究群扩张存在的刻画；. 3.研究抽象群上连通局部紧群拓扑与完全不连通局部紧群拓扑的刻画。

The study of locally compact groups begins from Sophus Lie's work in 19 century. Locally compact groups have a close connection with Number theory, Analysis, Representation theory and so on. Each locally compact group can be regarded as the extension of its connected component and the corresponding quotient group (i.e. totally disconnected locally compact group). On the other side, topological groups have a clsoe connection with order structure. All group topologies on an abstract group forms a lattice under inclusion. Therefore the study of topological and algebraic properties of topological groups from the perspective of order structure become a hot spot. We plan to study Gap problems of locally compact group topologies in the lattice of group topologies over a group in our project. The details of our study are as follows:.1.Study gaps in totally disconnected locally compact groups, connected locally compact groups, i.e.discuss about the existence and constructions of the predecessors and successors of the group topology in the lattice of all group topologies on a group..2.Study the characterizations of existence of topological group extension..3.Study the characterizations of connected (totally disconnected) locally compact group topology on an abstract group G?