A Riemann-Roch theorem asserts that some algebraically defined wrong– way map in K-theory agrees with a topologically defined one [BFM]. Bismut and Lott [BiLo] proved a Riemann–Roch theorem for smooth fiber bundles in which the topologically defined wrong–way map is the homotopy transfer of Becker–Gottlieb and Dold. We generalize their theorem, refine it, and prove a converse stating that an ap...

Oriented matroids have long been of use in various areas of combinatorics [BLS93]. Gelfand and MacPherson [GM92] initiated the use of oriented matroids in manifold and bundle theory, using them to formulate a combinatorial formula for the rational Pontrjagin classes of a differentiable manifold. MacPherson [Mac93] abstracted this into a manifold theory (combinatorial differential (CD) manifolds...

Abstract Narasihman and Ramanan proved in [Amer. J. Math. 83 (1961), 563–572] that an arbitrary connection a vector bundle over base space B can be obtained as the pull-back (via correctly chosen classifying map from into appropriate Grassmannian) of universal Grassmannian. The purpose this paper is to relate geometric properties pulled-back connection. More specifically, we describe conditions...

We prove that the order of canonical vector bundle over configuration space is $2$ for a general planar graph, and $4$ nonplanar graph.

The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization. 1 Big-isotropic structures Weak-Hamiltonian functions belong to the framewor...

The setting is the following: V is a real vector bundle on a compact space X, provided with a non degenerate quadratic form to which we associate a bundle of (real or complex) Clifford algebras denoted by C(V ); the quadratic form is implicit in this notation. We denote by M(V ) the Grothendieck group associated to the category of (real or complex) vector bundles provided with a structure of (t...

Abstract We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen–Macaulay (for short, aCM), and prove that projective space only smooth complete intersection with Picard rank one admits an aCM bundle. also obtain number results bundles over several specific varieties. As opposite situation we Torelli‐type prob...

Since the work of Crown [5] in the 1970’s, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset (omp) P(E). This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coecke [1], and the approach via decompositions of Harding [6]. Moreover, it provides a source of omps for t...

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bund...

Let X be a compact connected Riemann surface of genus g, where g ≥ 3. Denote by Mξ := M(n, ξ) the moduli space of stable vector bundles over X of rank n and fixed determinant ξ. If the degree deg(ξ) and the rank n are coprime, then there is a universal family of vector bundles, U , over X parametrized by Mξ. This family is unique up to tensoring by a line bundle that comes from Mξ. We fix one u...