Emery Thomas

Vector fields on manifolds

where n = dim M and 6» = ith Betti number of M ( = dim of Hi(M; Q)). Thus the geometric property of M having a nonzero vector field is expressed in terms of the algebraic invariant xM. We will discuss extensions of this idea to vector ^-fields, fields of ^-planes, and foliations of manifolds. All manifolds considered will be connected, smooth and without boundary; all maps will be continuous. For background information on manifolds and vector fields see [30], [34] and [67].

Complete solutions to a family of cubic Diophantine equations

Abstract The following theorem is proved: If n ≥ 1.365 × 107, then the Diophantine equation x 3 −(n−1)x 2 y−(n+2)xy 2 −y 3 =±1 has only the “trivial” solutions (±1,0), (0,±1), (±1,∓1) . Moreover, we show that for 0 ≤ n ≤ 1000, (∗) has non-trivial solutions if, and only if, n = 0, 1, 3. Finally, we conjecture that if n > 3, then (∗) has only trivial solutions.

ON FUNCTIONAL CUP-PRODUCTS AND THE TRANSGRESSION OPERATOR

Abstract : Let g be a map from a topological space X to a topological space Y. Suppose that X and Y are the total spaces of respective proper triads and that g is a triad map. It is then possible, in certain cases, to express the functional cupproduct in terms of an ordinary cup-product. This is applied to obtain a classical result about the Hopf invariant and to obtain some new information about the cohomology rings of certain classifying spaces. (Author)

ON THE COHOMOLOGY OF THE REAL GRASSMANN COMPLEXES AND THE CHARACTERISTIC CLASSES OF n-PLANE BUNDLES(')

1. Introduction. This paper deals with two things: first the cohomology of the real Grassmann spaces; and secondly, relations between the various characteristic classes of n-plane bundles. Let Rn be a real n-plane bundle over a paracompact space B. With Pn we associate the Stiefel-Whitney characteristic class

Homotopy-abelian Lie groups

where h* denotes the induced homomorphism. Note that #* is an isomorphism if A is a covering map and p, a ^ 2 . Hence if two topological groups have a common universal covering group then their higher homotopy groups are related by an isomorphism which is compatible with the Samelson product. Let <nrq(G), where g ^ l , denote the subset of w2q(G) consisting of elements ((3, fi), where fiCz7Tq(G). We assert the following

On the enumeration of cross-sections

THD PAPER is motivated by the desire to enumerate the classes of immersions of manifolds in euclidean spaces. There is no problem in the stable range: all immersions are equivalent by Whitney’s theorem. We concentrate on dimensions just out of this range. Hirsch [l] has reduced the problem to one of homotopy theory. In a preliminary note [3] we have reformulated the relevant part of his result so as to make it susceptible to the methods of Postnikov theory. Thus the immersion problem can be regarded as a special case of the following one.

Diophantine equations arising from cubic number fields

Abstract The solutions to a certain system of Diophantine equations and congruences determine, and are determined by, units in galois cubic number fields. These solutions fall into two classes: certain ones determine infinite families of solutions, while others do not. We construct an infinite number of examples of each type of solution. We obtain these results by relating certain pairs of units in arbitrary cubic number fields to solutions of a larger system of Diophantine equations.

Counting solutions to trinomial Thue equations: a different approach

We consider the problem of counting solutions to a trinomial Thue equation — that is, an equation |F (x, y)| = 1, (∗) where F is an irreducible form in Z[x, y] with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the “Thue-Siegel principle” and its relation to (∗). In this paper we give specific numerical bounds for the number of solutions to (∗) by a somewhat different approach, the difference lying in the initial step — solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.

Secondary Obstructions to Integrability

Publisher Summary This chapter discusses the problem that is if one has a smooth manifold M with a foliation F of codimension q then one associates with this a q -plane bundle η, the normal bundle to the foliation. Bott attacked this problem by showing if η ≈F ⊥ , then certain cup products of Pontryagin classes of η must vanish. This may be thought of as a first-lever result, in the sense that the cup product is a primary invariant. It is a general philosophy in algebraic topology that one should look for secondary invariants defined on the kernel of primary invariants, and this is precisely what Shulman does: he shows that if η ≈F ⊥ , then certain secondary products of Pontryagin classes of η (Massey products) also must vanish. Moreover, he shows the independence of his result from Botts by constructing a closed manifold with a bundle η such that the Bott invariants of η vanish but the Shulman invariants do not—in particular, then, η cannot be isomorphic to the normal bundle of a foliation. Having given rough outline of results, the chapter discusses them in detail, starting with the general definition of a Massey product.

Characteristic classes and differentiable manifolds

These lectures might be more accurately titled “The application of characteristic classes to geometric problems on manifolds.” In particular we will be interested in studying vector fields on manifolds. This first lecture gives the basic definitions we will need throughout the course.