Pub Date : 2020-03-03DOI: 10.1142/9789814667814_0037
L. Tu
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.
{"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}
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.
{"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}
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.
{"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}
Pub Date : 2020-03-03DOI: 10.23943/PRINCETON/9780691191751.003.0007
L. Tu
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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}
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