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Harmonic morphisms with one-dimensional fibres

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198503620
eISBN:
9780191708435
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198503620.003.0012
Subject:
Mathematics, Pure Mathematics

This chapter shows that a harmonic morphism from a manifold of dimension n+1 to a manifold of dimension n is, locally or globally, a principal bundle with a certain metric. When n = 3, in a ... More


Circle Actions

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0020
Subject:
Mathematics, Educational Mathematics

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 ... More


Free and Locally Free Actions

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0024
Subject:
Mathematics, Educational Mathematics

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 ... More


Proof of the Localization Formula for a Circle Action

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0031
Subject:
Mathematics, Educational Mathematics

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 ... More


Borel Localization for a Circle Action

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0026
Subject:
Mathematics, Educational Mathematics

This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated ... More


Introductory Lectures on Equivariant Cohomology: (AMS-204)

Loring W. Tu

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.001.0001
Subject:
Mathematics, Educational Mathematics

Equivariant cohomology is concerned with the algebraic topology of spaces with a group action, or in other words, with symmetries of spaces. First defined in the 1950s, it has been introduced into ... More


Localization Formulas

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0030
Subject:
Mathematics, Educational Mathematics

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. ... More


Integration of Equivariant Forms

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0028
Subject:
Mathematics, Educational Mathematics

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 ... More


Rationale for a Localization Formula

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0029
Subject:
Mathematics, Educational Mathematics

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 ... More


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