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Introductory Lectures on Equivariant Cohomology最新文献

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Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.45
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引用次数: 0
Overview 概述
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.23943/princeton/9780691191751.003.0001
L. Tu
{"title":"Overview","authors":"L. Tu","doi":"10.23943/princeton/9780691191751.003.0001","DOIUrl":"https://doi.org/10.23943/princeton/9780691191751.003.0001","url":null,"abstract":"This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121618597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some Applications 一些应用
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.38
L. Tu
{"title":"Some Applications","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.38","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.38","url":null,"abstract":"This chapter explores some applications of equivariant cohomology. Since its introduction in the Fifties, equivariant cohomology has found applications in topology, symplectic geometry, K-theory, and physics, among other fields. Its greatest utility may be in converting an integral on a manifold to a finite sum. Since many problems in mathematics can be expressed in terms of integrals, the equivariant localization formula provides a powerful computational tool. The chapter then discusses a few of the applications of the equivariant localization formula. In order to use the equivariant localization formula to compute the integral of an invariant form, the form must have an equivariantly closed extension.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"119 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141225348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Differential Graded Algebras 微分分级代数
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.1007/978-1-4614-8468-4_10
P. Griffiths, J. Morgan
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引用次数: 0
The Topology of a Group Action 组动作的拓扑结构
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.31
L. Tu
{"title":"The Topology of a Group Action","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.31","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.31","url":null,"abstract":"This chapter describes the topology of a group action. It proves some topological facts about the fixed point set and the stabilizers of a continuous or a smooth action. The chapter also introduces the equivariant tubular neighborhood theorem and the equivariant Mayer–Vietoris sequence. A tubular neighborhood of a submanifold S in a manifold M is a neighborhood that has the structure of a vector bundle over S. Because the total space of a vector bundle has the same homotopy type as the base space, in calculating cohomology one may replace a submanifold by a tubular neighborhood. The tubular neighborhood theorem guarantees the existence of a tubular neighborhood for a compact regular submanifold. The Mayer–Vietoris sequence is a powerful tool for calculating the cohomology of a union of two open subsets. Both the tubular neighborhood theorem and the Mayer–Vietoris sequence have equivariant counterparts for a G-manifold where G is a compact Lie group.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131424717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Curvature on a Principal Bundle 主束上的曲率
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.23
L. Tu
{"title":"Curvature on a Principal Bundle","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.23","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.23","url":null,"abstract":"This chapter examines curvature on a principal bundle. The curvature of a connection on a principal G-bundle is a g-valued 2-form that measures, in some sense, the deviation of the connection from the Maurer-Cartan connection on a product bundle. The Maurer-Cartan form Θ‎ on a Lie group G satisfies the Maurer-Cartan equation. Let M be a smooth manifold. The chapter then pulls the Maurer-Cartan equation back and uses Proposition 14.3 to get the Maurer-Cartan connection. It also considers the second structural equation; the first structural equation is discussed in a previous chapter. Finally, the chapter derives some properties of the curvature form.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"148 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120862610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Maurer–Cartan Form 毛雷尔-卡坦式
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.21
L. Tu
{"title":"The Maurer–Cartan Form","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.21","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.21","url":null,"abstract":"This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The chapter describes the Maurer-Cartan form and the equation it satisfies, the Maurer-Cartan equation. The Maurer-Cartan form allows one to define a connection on the product bundle M × G → M for any manifold M. The Lie algebra g of a Lie group G is defined to be the tangent space at the identity. One will often identify the two vector spaces and think of elements of g as left-invariant vector fields on G.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121728042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Borel Localization for a Circle Action 圆动作的Borel定位
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.32
L. Tu
{"title":"Borel Localization for a Circle Action","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.32","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.32","url":null,"abstract":"This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated on its fixed point set and that the isomorphism in localized equivariant cohomology of the manifold and its fixed point set is a ring isomorphism. This is clearly an important result in its own right. Moreover, since the fixed point set is a regular submanifold and is usually simpler than the manifold, the Borel localization theorem sometimes allows one to obtain the ring structure of the equivariant cohomology of an S1-manifold from that of its fixed point set. The chapter demonstrates this method with the example of S1 acting on S2 by rotations.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124028386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Lie Derivative and Interior Multiplication 李导数与内乘法
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.16
L. Tu
{"title":"The Lie Derivative and Interior Multiplication","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.16","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.16","url":null,"abstract":"This chapter reviews two operations on differential forms, the Lie derivative and interior multiplication. These are necessary to the definition of invariant forms, horizontal forms, and basic forms in the construction of the Cartan model. The chapter then looks at the Lie derivative of a vector field and of a differential form. The Lie derivative of a differential form is defined in a similar way to the Lie derivative of a vector field, but the chapter uses the pullback instead of the pushforward to compare nearby values. One can rearrange the product formula so that it becomes the global formula for the Lie derivative. Meanwhile, the interior multiplication is also called the contraction.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129931008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spectral Sequences 谱序列
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.12
L. Tu
{"title":"Spectral Sequences","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.12","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.12","url":null,"abstract":"This chapter focuses on spectral sequences. The spectral sequence is a powerful computational tool in the theory of fiber bundles. First introduced by Jean Leray in the 1940s, it was further refined by Jean-Louis Koszul, Henri Cartan, Jean-Pierre Serre, and many others. The chapter provides a short introduction, without proofs, to spectral sequences. As an example, it computes the cohomology of the complex projective plane. The chapter then details Leray's theorem. A spectral sequence is like a book with many pages. Each time one turns a page, one obtains a new page that is the cohomology of the previous page.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131800146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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