Torus Actions and their Applications in Topology and Combinatorics
Torus Actions and their Applications in Topology and Combinatorics
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DOI：
10.1090/ulect/024
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发表时间：
2002-04
期刊：
影响因子：
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通讯作者：
V. Buchstaber;T. Panov
中科院分区：
文献类型：
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作者：
V. Buchstaber;T. Panov
在这里，拓扑空间上的环面作用的研究，作为一个桥梁连接组合和凸几何与交换和同调代数，代数几何和拓扑。这个建立的联系有助于理解几何和拓扑空间与环面行动通过研究组合学的空间轨道。相反，组合对象的微妙性质可以通过将其解释为适当流形的轨道结构或作为由环面作用的复合体来实现。后者可以是具有Hamilton环面作用的辛流形、环面簇或流形、子空间排列补等，而组合对象包括单纯和立方复形、多面体和排列。这种方法还提供了一个自然的拓扑解释，在许多结构的环面行动，从交换和同调代数中使用的组合。围绕矩角复形理论展开论述，为用等变拓扑方法研究三角剖分不变量提供了一条有效途径。这本书包括许多新的和众所周知的开放问题，并将适合作为教科书。这将是有用的专家都在拓扑学和组合学，并将有助于建立更紧密的联系之间的主题。
Here, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This established link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, subtle properties of a combinatorial object can be realized by interpreting it as the orbit structure for a proper manifold or as a complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics. The exposition centers around the theory of moment-angle complexes, providing an effective way to study invariants of triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. It will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved.