Sum Of Bundles Research Articles

Invariant connections on a Whitney sum of bundles

Invariant connections on a Whitney sum of bundles

The Thom isomorphism for nonorientable bundles

The classical theory of Thom isomorphisms is extended to nonorientable vector bundles. The properties of orientation sheaves of bundles and of the Thom and Euler classes τ and e with respect to projections, fiber maps, Cartesian products, and Whitney sums of bundles are studied. The validity of standard constructions used in the applications of the classes τ and e is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality.

(0,2) CALABI–YAU SIGMA MODELS AND THEIR DUALITY

N=(0,2) heterotic string theories on Calabi–Yau spaces are considered based on linear sigma approaches. We construct (0,2) sigma models on complete intersection Calabi–Yau spaces, ℳ, realized in a product of weighted projective spaces. The definition of the associating vector bundle V over ℳ is given. These models are formulated as the IR limits of (0,2) U (1)N gauged linear sigma models. Examining their phase structures, we observe that there are hybrid phases where one cannot intrinsically distinguish between the defining polynomials of the Calabi–Yau space ℳ and those of the gauge bundle V. Thus we construct dual pairs of two (0,2) Calabi–Yau sigma models which are isomorphic in their hybrid phases. In addition, generalizing the gauge bundle into the direct sum of bundles, we study the duality of (0,2) string vacua.

Dimension and Stable Rank in the K -Theory of C* -Algebras

In topological K-theory, which can be viewed as the algebraic side of the theory of vector bundles, some of the interesting properties which one investigates are, for example, the conditions under which bundles must possess trivial direct summands, or the extent to which the cancellation property for direct (Whitney) sums of bundles holds. Such properties turn out to be controlled in part by the dimension of the base space, and results describing the nature of this control are among what are frequently called stability results [17]. During the past few years there has been a sudden flowering of a /^-theory for C*algebras. (See surveys [35, 36, 37].) Since C*-algebras are profitably thought of as 'non-commutative locally compact spaces', with the finitely generated projective modules being the appropriate generalization of vector bundles according to Swan's theorem [40], it would be natural to look for stability results for C*-algebras. But until now the theory has been focussed on the /C-groups of C*-algebras, and there has been little discussion of stability properties, presumably in part for lack of an appropriate concept of dimension for C*-algebras. One of the main objectives of this paper is to introduce a concept of dimension for C*-algebras which directly generalizes the classical concept of dimension for compact spaces [16, 20, 22], and to develop some techniques for calculating it, notably for C*algebras which are obtained as crossed-product algebras for an action of the group of integers. In algebraic K-theory, which is the generalization of topological /C-theory to rings, a substantial number of stability properties have been discovered [2, 39, 42-48]. Within that theory the concept which has played a role most analogous to that of dimension is the Bass stable rank. But despite its many successes, the Bass stable rank has often been difficult to calculate even for relatively uncomplicated rings. Another main objective of the present paper is to make accessible for application to C*algebras some of the stability results from algebraic /C-theory. This comes about from the fact that the concept of dimension which we introduce dominates the Bass stable rank. I was led to seek stability results for C*-algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation C*algebras discussed in [30] (which, as Elliott [11] has suggested, can appropriately be called 2-dimensional 'non-commutative tori'). In more concrete terms, the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent. The present paper provides the general theory which is needed for the affirmative answer which I obtained. In a subsequent paper I will derive the