发布求助
文献互助 智能选刊 最新文献

Introductory Lectures on Equivariant Cohomology最新文献

筛选
英文 中文
Hints and Solutions to Selected End-of-Section Problems 部分结尾问题的提示和解决方案
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.42
{"title":"Hints and Solutions to Selected End-of-Section Problems","authors":"","doi":"10.2307/j.ctvrdf1gz.42","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.42","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"19 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":"126794581","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
Connections on a Principal Bundle 主体包上的连接
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.1142/9789814667814_0037
L. Tu
{"title":"Connections on a Principal Bundle","authors":"L. Tu","doi":"10.1142/9789814667814_0037","DOIUrl":"https://doi.org/10.1142/9789814667814_0037","url":null,"abstract":"This chapter discusses connections on a principal bundle. Throughout the chapter, G will be a Lie group with Lie algebra g. One possible definition of a connection on a principal G-bundle P is a C∞ right-invariant horizontal distribution on P. Equivalently, a connection on P can be given by a right-equivariant g-valued 1-form on P that is the identity on vertical vectors. The chapter shows the equivalence of these two definitions of a connection. A connection is one of the most basic notions of differential geometry. It is essentially a way of differentiating sections. From a connection, the notions of curvature and geodesics follow.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"256 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":"114323980","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
Vector-Valued Forms 向量值形式
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.20
L. Tu
{"title":"Vector-Valued Forms","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.20","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.20","url":null,"abstract":"This chapter studies vector-valued forms. Ordinary differential forms have values in the field of real numbers. This chapter allows differential forms to take values in a vector space. When the vector space has a multiplication, for example, if it is a Lie algebra or a matrix group, the vector-valued forms will have a corresponding product. Vector-valued forms have become indispensable in differential geometry, since connections and curvature on a principal bundle are vector-valued forms. All the vector spaces will be real vector spaces. A k-covector on a vector space T is an alternating k-linear function. If V is another vector space, a V-valued k-covector on T is an alternating k-linear function.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"39 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":"132351939","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
Rationale for a Localization Formula 本地化公式的基本原理
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.35
L. Tu
{"title":"Rationale for a Localization Formula","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.35","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.35","url":null,"abstract":"This chapter offers a rationale for a localization formula. It looks at the equivariant localization formula of Atiyah–Bott and Berline–Vergne. The equivariant localization formula of Atiyah–Bott and Berline–Vergne expresses, for a torus action, the integral of an equivariantly closed form over a compact oriented manifold as a finite sum over the fixed point set. The central idea is to express a closed form as an exact form away from finitely many points. Throughout his career, Raoul Bott exploited this idea to prove many different localization formulas. The chapter then considers circle actions with finitely many fixed points. It also studies the spherical blow-up.","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":"132158198","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: 10.23943/PRINCETON/9780691191751.003.0007
L. Tu
{"title":"Equivariant Cohomology of S2 Under Rotation","authors":"L. Tu","doi":"10.23943/PRINCETON/9780691191751.003.0007","DOIUrl":"https://doi.org/10.23943/PRINCETON/9780691191751.003.0007","url":null,"abstract":"This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by rotation. The method of the chapter only gives the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient MG fits into Cartan's mixing diagram. One can then apply Leray's spectral sequence of the fiber bundle MG → BG to compute the equivariant cohomology from the cohomology of M and the cohomology of the classifying space BG.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"15 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":"115421087","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 of Equivariant Forms 等变形式的积分
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.34
L. Tu
{"title":"Integration of Equivariant Forms","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.34","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.34","url":null,"abstract":"This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with coefficients that are invariant forms on M. Such a form can be integrated by integrating the coefficients. This can be called equivariant integration. The chapter shows that under equivariant integration, Stokes's theorem still holds. Everything done so far in this book concerning a Lie group action on a manifold can be generalized to a manifold with boundary. An important fact concerning manifolds with boundary is that a diffeomorphism of a manifold with boundary takes interior points to interior points and boundary points to boundary points.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"68 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":"121800294","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
General Properties of Equivariant Cohomology 等变上同调的一般性质
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.15
L. Tu
{"title":"General Properties of Equivariant Cohomology","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.15","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.15","url":null,"abstract":"This chapter assesses the general properties of equivariant cohomology. Both the homotopy quotient and equivariant cohomology are functorial constructions. Equivariant cohomology is particularly simple when the action is free. Throughout the chapter, by a G-space, it means a left G-space. Let G be a topological group and consider the category of G-spaces and G-maps. A morphism of left G-spaces is a G-equivariant map (or G-map). Such a morphism induces a map of homotopy quotients. The map in turn induces a ring homomorphism in cohomology. The chapter then looks at the coefficient ring of equivariant cohomology, as well as the equivariant cohomology of a disjoint union.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"3 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":"127623314","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
Proof of the Localization Formula for a Circle Action 圆作用的定位公式的证明
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.37
L. Tu
{"title":"Proof of the Localization Formula for a Circle Action","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.37","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.37","url":null,"abstract":"This chapter provides a proof of the localization formula for a circle action. It evaluates the integral of an equivariantly closed form for a circle action by blowing up the fixed points. On the spherical blow-up, the induced action has no fixed points and is therefore locally free. The spherical blow-up is a manifold with a union of disjoint spheres as its boundary. For a locally free action, one can express an equivariantly closed form as an exact form. Since the localized equivariant cohomology of a locally free action is zero, after localization an equivariantly closed form must be equivariantly exact. Stokes's theorem then reduces the integral to a computation over spheres.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"188 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":"134143088","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
Acknowledgments 致谢
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.5
L. Tu
{"title":"Acknowledgments","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.5","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.5","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"6 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":"129399983","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
Basic Forms 基本形式
Introductory Lectures on Equivariant Cohomology Pub Date : 2020-03-03 DOI: 10.2307/j.ctvrdf1gz.18
L. Tu
{"title":"Basic Forms","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.18","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.18","url":null,"abstract":"This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"295 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":"122293258","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: 2
首页 上一页 下一页 尾页
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整，为了账户安全，请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
请完成安全验证×
相关产品
×
本文献相关产品
联系我们：info@booksci.cn Book学术提供免费学术资源搜索服务，方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号：481959085
Book学术官方微信