Dale Husemoller

Vector Fields on the Sphere

In Chap. 12, Theorem (8.2), we saw that S n −1 has ρ(n) − 1 orthonormal tangent vector fields defined on it. The object of this chapter is to outline the steps required to prove that S n −1 does not have ρ(n) orthonormal tangent vector fields defined on it; in fact, S n −1 does not have ρ(n) linearly independent tangent vector fields; see also Adams [6].

Homology of Certain H-Spaces as Group Ring Objects

Publisher Summary This chapter discusses homology of certain H -spaces as group ring objects. The chapter focuses on a group ring object for connected groups in the category of commutative coalgebras, that is, co-commutative Hopf algebras. A ring object in a category with finite products is triple where the pair is a commutative group object and the operation is associative. A connected ring can be defined as a ring with unit such that it induces an isomorphism in degree zero. There are two preliminaries needed for the construction of the group ring object. The chapter also presents the basic classes of examples of group rings.

Change of Structure Group in Fibre Bundles

In this chapter we consider the relation between principal H-bundles and principal G-bundles, where H is a closed subgroup of G. We do this for general principal bundles and then describe the relation, using the classifying spaces and the local coordinate description. This is a generalization of the process in Chap. 5, Sec. 7.

Characteristic Classes of Manifolds

for a closed manifold M is an isomorphism called Poincaré duality. This internal symmetry between homology and cohomology is a fundamental property of manifolds. When the manifold has a boundary ∂M or when it is not compact, then Poincaré duality must be modified, but it is always given by cap product with the fundamental class in Hn(M,Z) for the oriented case and in Hn(M,Z/2) for the general case. A smooth manifold M of dimension n has two complementary real vector bundles: the tangent bundle T (M) of dimension n and the normal bundle ν(M) to some embedding of M into Euclidean space. The Whitney sum T (M)⊕ν(M) is a trivial bundle on M which is the restriction to M of the trivial tangent bundle on Euclidean space. For the characteristic classes with the Whitney sum property, it is essentially equivalent to work either the characteristic classes of the tangent bundle or the normal bundle. Polynomial combinations of characteristic classes of the tangent bundle can be evaluated on the fundamental class [M], and the result is characteristic numbers which can be related to other invariants of the manifold. These numbers are related to homological invariants of the manifold and in many cases to other geometric invariants of the manifold and bundles on the manifold. In the last section, we illustrate this with an explication of the Riemann-Roch-Hirzebruch theorem. Chapter 18 of Fibre Bundles (Husemöller 1994) is a reference for this chapter.

Stability Properties of Vector Bundles

Two vector bundles ξ and η are called s-equivalent provided ξ ⨁θ n and η ⨁θ m are isomorphic for some n and m where θ m denotes the m dimensional trivial vector bundle. Stable equivalence, or s-equivalence, is an equivalence relation, and the stable classes form a ring (over finite-dimensional spaces), with ⨁ inducing the addition operation and ⨂ the multiplication operation. These are the \(\tilde K\)-rings of the space. We study the relation between isomorphism and stable equivalence. Also we consider elementary properties of the cofunctor \(\tilde K\).

The Gauge Group of a Principal Bundle

The gauge group of a principal bundle is simply its automorphism group with a topology coming from the mapping space topology. The mapping space topology (called the compact open topology) will not be considered in detail but will be outlined in the first section.

Characteristic Classes and Connections

Apart from the previous chapter, the theory of fibre bundles in this book is a theory over an arbitrary space. Even the relation to manifolds in Chapter 18 is treated from a topological point of view, but in the context of smooth manifolds and vector bundles we can approach Chern classes using constructions from analysis. This idea, which goes back to a letter from A. Weil (see A. Weil Collected papers, Volume III, pages 422–36 and 571–574), involves choosing a connection or covariant derivative on the complex vector bundle, defining the curvature 2-form of the connection, and representing the characteristic class as closed 2q-form which is a polynomial in the curvature form. This proceedure is outlined in this chapter.

General Fibre Bundles

A fibre bundle is a bundle with an additional structure derived from the action of a topological group on the fibres. In the next chapter the notions of fibre bundle and vector bundle are related. As with vector bundles, a fibre bundle has so much structure that there is a homotopy classification theorem for fibre bundles.

Elliptic Curves over Local Fields

We return to the ideas of Chapter 5 where the torsion in E(Q) was studied using the reduction map E(Q) → Ē(F p ). These local questions are taken up for any complete field K with a discrete valuation where the congruences of 5(4.3) are interpreted using the formal group introduced in Chapter 12, §7. This leads to a more precise version of 5(4.5).

L-Function of an Elliptic Curve and Its Analytic Continuation

We introduce the L-function of an elliptic curve E over a number field and derive its elementary convergence properties. An L-function of this type was first introduced by Hasse, and the concept was greatly extended by Weil. For this reason it is frequently called the Hasse-Weil L-function.