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Connections, Curvature, and Cohomology

This entry is about the book

• Werner Greub, Stephen Halperin, Ray Vanstone

  Connections, Curvature, and Cohomology

  Academic Press (1973)

on Chern-Weil theory: principal bundles with connections and their characteristic classes.

Related books are

• Theodore Frankel, The Geometry of Physics - An Introduction

Volume I

0 Algebraic and analytic preliminaries

• linear algebra

• homological algebra

• analysis, topology

1 Basic concepts

• topological manifold

• smooth manifold

• fiber bundle

II Vector bundles

• vector bundle

• section

III Tangent bundle and differential forms

• tangent bundle

• vector field

• differential form

• orientation

IV Calculus of differential forms

• Cartan calculus

• integration

• Stokes theorem

V De Rham cohomology

• de Rham cohomology

• Poincare duality

• Künneth theorem

• de Rham theorem

VI Mapping degree

VII Integration over the fiber

• fiber integration

VIII Cohomology of sphere bundles

• Euler class

IX Cohomology of vector bundles

• Thom isomorphism

X The Lefschetz class of a manifold

• Lefschetz isomorphism?

Appendix A The exponential map

Volume II

0 Algebraic and analytic preliminaries

I Lie groups

• Lie group

II Subgroups and homogeneous spaces

• homogeneous space

III Transformation groups

IV Invariant cohomology

• equivariant cohomology

V Bundles with structrue group

• principal bundle

VI Principal connections and the Weil homomorphism

• connection on a bundle

• Chern-Weil homomorphism

VII Linear connections

VIII Characteristic homomorphism for $\Sigma$-bundles

IX Pontrjagin, Pfaffian, Chern classes

• Pontryagin class

• Pfaffian

• Chern class

X The Gauss-Bonnet-Chern theorem

• Gauss-Bonnet theorem

Appendix A Characteristic coefficients and the Pfaffian

Volume III

0 Algebraic preliminaries

I Spectral sequences

• fiber sequence

• spectral sequence

II Koszul complexes of $P$-spaces and $P$-algebras

III Koszul complexes of $P$-differential algebras

IV Lie algebras and differential spaces

• Lie algebra

V Cohomology of Lie algebras and Lie groups

• Lie algebra cohomology

• Lie group cohomology

VI The Weil alebra

• Weil algebra

VII Operation of a Lie algebra in a graded differential algebra

VIII Algebraic connections and principal bundles

IX Cohomology of operations and principal bundles

X Subalgebras

XI Homogeneous spaces

XII Operation of a Lie algebra on a pair

Appendix A Characteristic coefficients and the Pfaffian

category: reference