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Introductory Lectures on Equivariant Cohomology最新文献

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Universal Bundles and Classifying Spaces 泛束与分类空间
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.23943/PRINCETON/9780691191751.003.0005
L. Tu
{"title":"Universal Bundles and Classifying Spaces","authors":"L. Tu","doi":"10.23943/PRINCETON/9780691191751.003.0005","DOIUrl":"https://doi.org/10.23943/PRINCETON/9780691191751.003.0005","url":null,"abstract":"This chapter evaluates universal bundles and classifying spaces. As before, G is a topological group. In defining the equivariant cohomology of a G-space M, one needs a weakly contractible space EG on which G acts freely. Such a space is provided by the total space of a universal G-bundle, a bundle from which every principal G-bundle can be pulled back. The base BG of a universal G-bundle is called a classifying space for G. By Whitehead's theorem, for CW-complexes, weakly contractible is the same as contractible. In the category of CW complexes (with continuous maps as morphisms), a principal G-bundle whose total space is contractible turns out to be precisely a universal G-bundle.","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":"129049855","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
Fundamental Vector Fields 基本向量场
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.17
L. Tu
{"title":"Fundamental Vector Fields","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.17","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.17","url":null,"abstract":"This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"22 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":"130680517","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
Integration on a Compact Connected Lie Group 紧连通李群上的积分
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.19
L. Tu
{"title":"Integration on a Compact Connected Lie Group","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.19","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.19","url":null,"abstract":"This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"25 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":"133328707","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
A Crash Course in Representation Theory 表征理论速成班
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.33
L. Tu
{"title":"A Crash Course in Representation Theory","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.33","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.33","url":null,"abstract":"This chapter studies representation theory. In order to state the equivariant localization formula of Atiyah–Bott and Berline–Vergne, one will need to know some representation theory. Representation theory “represents” the elements of a group by matrices in such a way that group multiplication becomes matrix multiplication. It is a way of simplifying group theory. The chapter provides the minimal representation theory needed for equivariant cohomology. A real representation of a group G is a group homomorphism. Every representation has at least two invariant subspaces, 0 and V. These are called the trivial invariant subspaces. A representation is said to be irreducible if it has no invariant subspaces other than 0 and V; otherwise, it is reducible.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"70 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":"127839157","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
Homotopy Quotients and Equivariant Cohomology 同伦商与等变上同调
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.10
L. Tu
{"title":"Homotopy Quotients and Equivariant Cohomology","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.10","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.10","url":null,"abstract":"This chapter investigates two candidates for equivariant cohomology and explains why it settles on the Borel construction, also called Cartan's mixing construction. Let G be a topological group and M a left G-space. The Borel construction mixes the weakly contractible total space of a principal bundle with the G-space M to produce a homotopy quotient of M. Equivariant cohomology is the cohomology of the homotopy quotient. More generally, given a G-space M, Cartan's mixing construction turns a principal bundle with fiber G into a fiber bundle with fiber M. Cartan's mixing construction fits into the Cartan's mixing diagram, a powerful tool for dealing with equivariant cohomology.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"42 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":"126335963","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
List of Figures 数字一览表
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.3
{"title":"List of Figures","authors":"","doi":"10.2307/j.ctvrdf1gz.3","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.3","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"4 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":"128295714","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: 10.23943/PRINCETON/9780691191751.003.0018
L. Tu
{"title":"Differential Graded Algebras","authors":"L. Tu","doi":"10.23943/PRINCETON/9780691191751.003.0018","DOIUrl":"https://doi.org/10.23943/PRINCETON/9780691191751.003.0018","url":null,"abstract":"This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"125 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":"130725349","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
Equivariant Cohomology of 𝑆² Under Rotation 旋转下𝑆²的等变上同调
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.13
{"title":"Equivariant Cohomology of 𝑆² Under Rotation","authors":"","doi":"10.2307/j.ctvrdf1gz.13","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.13","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"11 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":"125744050","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
Universal Bundles and Classifying Spaces 泛束与分类空间
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.1007/BFB0096870
A. Borel
{"title":"Universal Bundles and Classifying Spaces","authors":"A. Borel","doi":"10.1007/BFB0096870","DOIUrl":"https://doi.org/10.1007/BFB0096870","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"65 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":"125966036","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
A Universal Bundle for a Compact Lie Group 紧李群的全称束
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.14
L. Tu
{"title":"A Universal Bundle for a Compact Lie Group","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.14","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.14","url":null,"abstract":"This chapter looks at a universal bundle for a compact Lie group. By Milnor's construction, every topological group has a universal bundle. Independently of Milnor's result, the chapter constructs a universal bundle for any compact Lie group G. This construction is based on the fact that every compact Lie group can be embedded as a subgroup of some orthogonal group O(k). The chapter first constructs a universal O(k)-bundle by finding a weakly contractible space on which O(k) acts freely. The infinite Stiefel variety V (k, ∞) is such a space. As a subgroup of O(k), the compact Lie group G will also act freely on V (k, ∞), thereby producing a universal G-bundle.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"72 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":"121697976","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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