Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

Circle Actions

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.

Integration of Equivariant Forms

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.

Free and Locally 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.

The Cartan Model in General

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.

The Lie Derivative and Interior Multiplication

This chapter reviews two operations on differential forms, the Lie derivative and interior multiplication. These are necessary to the definition of invariant forms, horizontal forms, and basic forms in the construction of the Cartan model. The chapter then looks at the Lie derivative of a vector field and of a differential form. The Lie derivative of a differential form is defined in a similar way to the Lie derivative of a vector field, but the chapter uses the pullback instead of the pushforward to compare nearby values. One can rearrange the product formula so that it becomes the global formula for the Lie derivative. Meanwhile, the interior multiplication is also called the contraction.

Thom form in equivariant Čech–de Rham theory

In the present paper, we provide the foundation of a [Formula: see text]-equivariant Čech–de Rham theory for a compact Lie group [Formula: see text] by using the Cartan model of equivariant differential forms. Our approach is quite elementary without referring to the Mathai–Quillen framework. In particular, by a direct computation, we give an explicit formula of the [Formula: see text]-equivariant Thom form of [Formula: see text], which deforms the classical Bochnor–Martinelli kernel. Also, we discuss a version of equivariant Riemann–Roch formula.