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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 neighbourhood of a critical point, a harmonic morphism behaves like the Hopf polynomial map; when n > 3, there can be no critical points. A factorization theorem and a circle action are obtained in all cases, leading to topological restrictions. Given a nowhere-zero Killing field V, it is shown how to find harmonic morphisms with fibres tangent to V. Harmonic morphisms of warped product type are discussed; these are related to isoparametric functions. These two types are the only types that can occur on a space form or on an Einstein manifold when n > 3. When n = 3, a third type of harmonic morphism is found related to the Beltrami fields equation of hydrodynamics.

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.

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.

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 spherical blow-up, the induced action has no fixed points and is therefore locally free. The spherical blow-up is a manifold with a union of disjoint spheres as its boundary. For a locally free action, one can express an equivariantly closed form as an exact form. Since the localized equivariant cohomology of a locally free action is zero, after localization an equivariantly closed form must be equivariantly exact. Stokes's theorem then reduces the integral to a computation over spheres.

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 on its fixed point set and that the isomorphism in localized equivariant cohomology of the manifold and its fixed point set is a ring isomorphism. This is clearly an important result in its own right. Moreover, since the fixed point set is a regular submanifold and is usually simpler than the manifold, the Borel localization theorem sometimes allows one to obtain the ring structure of the equivariant cohomology of an S¹-manifold from that of its fixed point set. The chapter demonstrates this method with the example of S¹ acting on S² by rotations.

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.

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.

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 Berline–Vergne expresses, for a torus action, the integral of an equivariantly closed form over a compact oriented manifold as a finite sum over the fixed point set. The central idea is to express a closed form as an exact form away from finitely many points. Throughout his career, Raoul Bott exploited this idea to prove many different localization formulas. The chapter then considers circle actions with finitely many fixed points. It also studies the spherical blow-up.