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.
{"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}
Pub Date : 2020-03-03DOI: 10.23943/PRINCETON/9780691191751.003.0005
L. Tu
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.
{"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}
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.
{"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}
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.
{"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}
{"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}
Pub Date : 2020-03-03DOI: 10.23943/PRINCETON/9780691191751.003.0018
L. Tu
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.
{"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}
{"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}
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.
{"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}
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