Homotopy Quotients and Equivariant Cohomology

2020 ◽  
pp. 29-38
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Fiber Bundle ◽  
Equivariant Cohomology ◽  
Principal Bundle ◽  
Total Space

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.

Principal Bundles

2020 ◽  
pp. 21-28
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Fiber Bundle ◽  
Equivariant Cohomology ◽  
Special Kind ◽  
Total Space ◽  
Principal Bundles ◽  
The Right

This chapter examines principal bundles. Throughout the chapter, G will be a topological group. It then defines a principal G-bundle and provides a criterion for a map to be a principal G-bundle. This is followed by several examples of principal G-bundles. A principal G-bundle is a special kind of fiber bundle with fiber G such that the group G acts freely on the right on the total space of the bundle. Equivariant cohomology is defined in terms of a special principal G-bundle whose total space is weakly contractible. Such a bundle turns out to be what can be called a universal principal G-bundle.

Equivariant Cohomology of S2 Under Rotation

2020 ◽  
pp. 57-60
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Spectral Sequence ◽  
Fiber Bundle ◽  
Equivariant Cohomology ◽  
Module Structure ◽  
Classifying Space

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.

Universal Bundles and Classifying Spaces

2020 ◽  
pp. 39-44
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Equivariant Cohomology ◽  
Total Space ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Continuous Maps ◽  
Cw Complexes

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.

scholarly journals SCATTERING, GREEN’S FUNCTIONS AND SYMMETRIES IN A TOTAL SPACE OF A DIRAC MONOPOLE FIBRE BUNDLE

1991 ◽  
Vol 06 (04) ◽  
pp. 577-598 ◽  
Author(s):  
A.G. SAVINKOV ◽  
A.B. RYZHOV
Keyword(s):  
Green's Functions ◽  
Principal Bundle ◽  
Fibre Bundle ◽  
Green’S Functions ◽  
Wave Functions ◽  
Total Space ◽  
Dirac Monopole ◽  
Hidden Symmetries ◽  
Scattering Wave ◽  
A Charge

The scattering wave functions and Green’s functions were found in a total space of a Dirac monopole principal bundle. Also, hidden symmetries of a charge-Dirac monopole system and those joining the states relating to different topological charges n=2eg were found.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

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.

scholarly journals Dispersive semiflows on fiber bundles

2017 ◽  
Vol 37 (2) ◽  
pp. 85-99
Author(s):  
Josiney A. Souza ◽  
Hélio V. M. Tozatti
Keyword(s):  
Periodic Point ◽  
Fiber Bundle ◽  
Vector Bundles ◽  
Base Space ◽  
Total Space ◽  
Fiber Bundles ◽  
Almost Periodic ◽  
Special Result ◽  
Almost Periodic Point

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.

General Properties of Equivariant Cohomology

2020 ◽  
pp. 69-76
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Equivariant Cohomology ◽  
Disjoint Union ◽  
Ring Homomorphism ◽  
Equivariant Map ◽  
General Properties ◽  
Coefficient Ring

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.

Topological aspects of the projective unitary group

1970 ◽  
Vol 68 (1) ◽  
pp. 57-60 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Hilbert Space ◽  
Topological Group ◽  
Projective Space ◽  
Unitary Group ◽  
Principal Bundle ◽  
Strong Operator Topology ◽  
Complex Hilbert Space ◽  
Unitary Representations ◽  
Strong Operator ◽  
Unitary Transformations

1. Introduction. The group U(H) of unitary transformations of a complex Hilbert space H, endowed with its strong operator topology, is of interest in the study of unitary representations of a topological group. The unitary transformations of H induce a group U(Ĥ) of transformations of the associated projective space Ĥ. The projective unitary group U(Ĥ) with its strong operator topology is used in the study of projective (ray) representations. U(Ĥ) is, as a group, the quotient of U(H) by the subgroup S1 of scalar multiples of the identity. In this paper we prove that the strong operator toplogy of U(Ĥ) is in fact the quotient of the strong operator topology on U(H). This is related to the fact that U(H) is a principal bundle over U(Ĥ) with fibre S.

Free and Locally Free Actions

2020 ◽  
pp. 197-204
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Equivariant Cohomology ◽  
Identity Element ◽  
Circle Action ◽  
Cartan Model ◽  
Free Actions

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.

scholarly journals The whitehead torsion of the total space of a fiber bundle

Topology ◽  
1972 ◽  
Vol 11 (2) ◽  
pp. 179-194 ◽  
Author(s):  
Douglas R. Anderson
Keyword(s):  
Fiber Bundle ◽  
Total Space
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