This chapter addresses free and locally free actions. It uses the Cartan model to compute the equivariant cohomology of a circle action, so equivariant cohomology is taken with real coefficients. An action is said to be free if the stabilizer of every point consists only of the identity element. It turns out that the equivariant cohomology of a free circle action is always u-torsion. More generally, an action of a topological group G on a topological space X is locally free if the stabilizer Stab(x) of every point is discrete. The chapter then proves that the equivariant cohomology of a locally free circle action on a manifold is also u-torsion.
{"title":"Free and Locally Free Actions","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.30","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.30","url":null,"abstract":"This chapter addresses free and locally free actions. It uses the Cartan model to compute the equivariant cohomology of a circle action, so equivariant cohomology is taken with real coefficients. An action is said to be free if the stabilizer of every point consists only of the identity element. It turns out that the equivariant cohomology of a free circle action is always u-torsion. More generally, an action of a topological group G on a topological space X is locally free if the stabilizer Stab(x) of every point is discrete. The chapter then proves that the equivariant cohomology of a locally free circle action on a manifold is also u-torsion.","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":"122414558","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 an outline of a proof of the equivariant de Rham theorem. In 1950, Henri Cartan proved that the cohomology of the base of a principal G-bundle for a connected Lie group G can be computed from the Weil model of the total space. From Cartan's theorem it is not too difficult to deduce the equivariant de Rham theorem for a free action. Guillemin and Sternberg presents an algebraic proof of the equivariant de Rham theorem, although some details appear to be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of a different approach using the Mayer–Vietoris argument. A limitation of the Mayer–Vietoris argument is that it applies only to manifolds with a finite good cover. The chapter provides a proof of the general case with no restrictions on the manifold and with all the details.
{"title":"Outline of a Proof of the Equivariant de Rham Theorem","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.28","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.28","url":null,"abstract":"This chapter offers an outline of a proof of the equivariant de Rham theorem. In 1950, Henri Cartan proved that the cohomology of the base of a principal G-bundle for a connected Lie group G can be computed from the Weil model of the total space. From Cartan's theorem it is not too difficult to deduce the equivariant de Rham theorem for a free action. Guillemin and Sternberg presents an algebraic proof of the equivariant de Rham theorem, although some details appear to be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of be missing. Guillemin, Ginzburg, and Karshon outline in an appendix of a different approach using the Mayer–Vietoris argument. A limitation of the Mayer–Vietoris argument is that it applies only to manifolds with a finite good cover. The chapter provides a proof of the general case with no restrictions on the manifold and with all the details.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"40 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":"132568812","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 digression concerning the all-important technique of localization in algebra. Localization generally means formally inverting a multiplicatively closed subset in a ring. However, the chapter focuses on the particular case of inverting all nonnegative powers of a variable u in an ℝ[u]-module. Localization of an ℝ[u]-module with respect to a variable u kills the torsion elements and preserves exactness. The chapter then looks at the proposition that localization preserves the direct sum. The simplest proof for this proposition is probably one that uses the universal mapping property of the direct sum. The chapter also considers antiderivations under localization.
{"title":"Localization in Algebra","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.29","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.29","url":null,"abstract":"This chapter provides a digression concerning the all-important technique of localization in algebra. Localization generally means formally inverting a multiplicatively closed subset in a ring. However, the chapter focuses on the particular case of inverting all nonnegative powers of a variable u in an ℝ[u]-module. Localization of an ℝ[u]-module with respect to a variable u kills the torsion elements and preserves exactness. The chapter then looks at the proposition that localization preserves the direct sum. The simplest proof for this proposition is probably one that uses the universal mapping property of the direct sum. The chapter also considers antiderivations under localization.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"764 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":"133152425","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 focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.
{"title":"Circle Actions","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.26","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.26","url":null,"abstract":"This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"85 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":"114735162","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 the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because the condition that LX α = 0 is sufficient for a differential form α on M to be invariant holds only for a connected Lie group. It also considers the theorem that marks the transition from the Weil model to the Cartan model. It is due to Henri Cartan, who played a crucial role in the development of equivariant cohomology. The chapter then studies the Weil-Cartan isomorphism.
{"title":"The Cartan Model in General","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.27","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.27","url":null,"abstract":"This chapter looks at the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because the condition that LX α = 0 is sufficient for a differential form α on M to be invariant holds only for a connected Lie group. It also considers the theorem that marks the transition from the Weil model to the Cartan model. It is due to Henri Cartan, who played a crucial role in the development of equivariant cohomology. The chapter then studies the Weil-Cartan isomorphism.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"54 8-9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120921224","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 discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.
{"title":"Homotopy Groups and CW Complexes","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.8","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.8","url":null,"abstract":"This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.","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":"117314173","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 evaluates the Weil algebra and the Weil model. The Weil algebra of a Lie algebra g is a g-differential graded algebra that in a definite sense models the total space EG of a universal bundle when g is the Lie algebra of a Lie group G. The Weil algebra of the Lie algebra g and the map f is called the Weil map. The Weil map f is a graded-algebra homomorphism. The chapter then shows that the Weil algebra W(g) is a g-differential graded algebra. The chapter then looks at the cohomology of the Weil algebra; studies algebraic models for the universal bundle and the homotopy quotient; and considers the functoriality of the Weil model.
{"title":"The Weil Algebra and the Weil Model","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.25","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.25","url":null,"abstract":"This chapter evaluates the Weil algebra and the Weil model. The Weil algebra of a Lie algebra g is a g-differential graded algebra that in a definite sense models the total space EG of a universal bundle when g is the Lie algebra of a Lie group G. The Weil algebra of the Lie algebra g and the map f is called the Weil map. The Weil map f is a graded-algebra homomorphism. The chapter then shows that the Weil algebra W(g) is a g-differential graded algebra. The chapter then looks at the cohomology of the Weil algebra; studies algebraic models for the universal bundle and the homotopy quotient; and considers the functoriality of the Weil model.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"34 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":"125683223","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 highlights localization formulas. The equivariant localization formula for a torus action expresses the integral of an equivariantly closed form as a finite sum over the fixed point set. It was discovered independently by Atiyah and Bott on the one hand, and Berline and Vergne on the other, around 1982. The chapter describes the equivariant localization formula for a circle action and works out an application to the surface area of a sphere. It also explores some equivariant characteristic classes of a vector bundle. These include the equivariant Euler class, the equivariant Pontrjagin classes, and the equivariant Chern classes.
{"title":"Localization Formulas","authors":"L. Tu","doi":"10.2307/j.ctvrdf1gz.36","DOIUrl":"https://doi.org/10.2307/j.ctvrdf1gz.36","url":null,"abstract":"This chapter highlights localization formulas. The equivariant localization formula for a torus action expresses the integral of an equivariantly closed form as a finite sum over the fixed point set. It was discovered independently by Atiyah and Bott on the one hand, and Berline and Vergne on the other, around 1982. The chapter describes the equivariant localization formula for a circle action and works out an application to the surface area of a sphere. It also explores some equivariant characteristic classes of a vector bundle. These include the equivariant Euler class, the equivariant Pontrjagin classes, and the equivariant Chern classes.","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"1998 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133003464","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 : 1995-12-26DOI: 10.1201/NOE0849324796.BMATT
D. Zwillinger
{"title":"List of Notations","authors":"D. Zwillinger","doi":"10.1201/NOE0849324796.BMATT","DOIUrl":"https://doi.org/10.1201/NOE0849324796.BMATT","url":null,"abstract":"","PeriodicalId":272846,"journal":{"name":"Introductory Lectures on Equivariant Cohomology","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1995-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124739873","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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