Construction of the topological toric theory

我们从拓扑学的角度发展了环面变分理论。这不仅从拓扑学的角度对环面变异理论进行了重构，而且提供了多扇形、多多面体等新的研究对象，使本研究更加有趣。与环面流形相对应的拓扑对象是环面流形。我们的研究目的是研究环面流形的几何性质。(1)具有消失奇次上同调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。我在和Panov的合作中用轨道空间描述了那些环面流形。在这项研究的激励下，我对球体的简单细胞分解的数量进行了表征。这解决了Stanley的一个猜想。(2) Guillemin-Zara研究了环面流形拓扑与图论的关系。这在图论中引入了等变拓扑的思想，非常有趣。证明了具有轴函数的环面图的等变上同环与一个简单偏序集的面环是一致的。(3)小覆盖的概念在某种意义上是环面理论的真实版本。这与环面理论有很多相似之处，但也有一些本质上的区别，例如，一个小洞有一个非平凡的基本群，大多数小洞是不可定向的，而每个环面流形都是单连通和可定向的。我尝试对立方体上的小覆盖物以及立方体上的圆环流形进行分类。

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.