Construction of the topological toric theory

我们从拓扑学的角度发展了环簇理论。这不仅是对环簇理论在拓扑学上的重构，而且提供了多扇、多多面体等新的研究对象，使本文的研究更具趣味性。与环面流形对应的拓扑物是环面流形。研究了环面流形的几何性质。(1)奇次上同调为零的环面流形在环面流形中表现良好。事实上，环面流形具有这样的性质。在与帕诺夫的合作中，我已经根据轨道空间刻画了这些环面流形的特征。在这项研究的启发下，我刻画了球面单纯细胞分解的单纯形数。这解决了Stanley的一个猜想。(2)Guillmin-Zara研究了环面流形的拓扑与图论之间的关系。这在图论中引入了等变拓扑的概念，非常有趣。我们证明了具有轴函数的环面图的等变上同调环与单纯偏序集的面环一致。(3)小覆盖的概念在某种意义上是环面理论的实版。这与环面理论有许多相似之处，但也有一些本质的不同，例如，一个小的cever有一个非平凡的基本群，大多数小的洞穴是非对偶的，而每个环面流形都是单连通的和可定向的。我试着对立方体上的小盖子和立方体上的环形流形进行分类。

We have developed the theory of toric varieties from a topological point of view. This is not only reconstruction of the theory of toric varieties in terms of topology but also provides new objected such as multi-fan and multi-polytope etc.which makes this research more interesting. The topological object corresponding to toric manifold is a torus manifold. The purpose of our research was to study geometrical properties of torus manifolds.(1) Torus manifolds which have vanishing odd degree cohomology behave well among torus manifolds. In fact, toric manifolds have such property. I have characterized those torus manifolds in terms of orbit space in a joint work with Panov. Motivated by this research, I have characterized the numbers of simplices of simplicial cell decompositions of spheres. This solves a conjecture by Stanley.(2) The relation between the topology of torus manifolds and graph theory is studied by Guillemin-Zara. This introduces an idea of equivariant topology in the theory of graph and is quite interesting. We have proved that the equivariant cohomology ring of a torus graph with axial function agrees with the face ring of a simplicial poset.(3) The notion of small cover is in some sense a real version of toric theory. This has a lot of similarity to toric theory but there are some essential differences For instance, a small cever has a non-trivial fundamental group and most of small cavers are non-onientable while every toric manifold is simply connected and orientable. I tried to classify small covers over cubes as well as toric manifold over cubes.