Oriented Vector Bundle Research Articles

The Kudla–Millson form via the Mathai–Quillen formalism

AbstractA crucial ingredient in the theory of theta liftings of Kudla and Millson is the construction of a $q$ -form $\varphi_{KM}$ on an orthogonal symmetric space, using Howe's differential operators. This form can be seen as a Thom form of a real oriented vector bundle. We show that the Kudla-Millson form can be recovered from a canonical construction of Mathai and Quillen. A similar result was obtaind by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to arbitrary signature.

Thom isomorphisms in triangulated motivic categories

We show that a triangulated motivic category admits categorical Thom isomorphisms for vector bundles with an additional structure if and only if the generalized motivic cohomology theory represented by the tensor unit object admits Thom classes. We also show that the stable $\mathbb{A}^1$-derived category does not admit Thom isomorphisms for oriented vector bundles and, more generally, for symplectic bundles. In order to do so we compute the first homology sheaves of the motivic sphere spectrum and show that the class in the coefficient ring of $\mathbb{A}^1$-homology corresponding to the second motivic Hopf map $\nu$ is nonzero which provides an obstruction to the existence of a reasonable theory of Thom classes in $\mathbb{A}^1$-cohomology.

Horizontality in the twistor spaces associated with vector bundles of rank 4 on tori

Let E be an oriented vector bundle of rank 4 over a torus $$T^2$$ with a metric h and $$\hat{E}$$ the twistor space associated with E. We will study the horizontality in $$\hat{E}$$ with respect to the connection $$\hat{\nabla }$$ induced by an h-connection $$\nabla $$ . We will describe the subset of the fiber $$\hat{E}_a$$ of $$\hat{E}$$ at a point $$a\in T^2$$ given by horizontality along normal polygonal curves in $${{\varvec{R}}}^2$$ for an initial value at a in the case where the subset is finite. We will see that if $$\hat{E}$$ has a partially horizontal section $$\Omega $$ , then there exists an h-connection $$\nabla '$$ related to $$\nabla $$ such that h, $$\nabla '$$ and $$\Omega $$ give a Kahler structure of E. We will make analogous discussions for an oriented vector bundle of rank 4 over $$T^2$$ with a neutral metric.

Euler classes of vector bundles over manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

Chow–Witt rings of classifying spaces for symplectic and special linear groups

We compute the Chow–Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with I j -coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. The computations for the symplectic groups show that Chow–Witt groups are a symplectically oriented ring cohomology theory. Using our computations for special linear groups, we also discuss the question when an oriented vector bundle of odd-rank splits off a trivial summand.

Characteristic rank of canonical vector bundles over oriented Grassmann manifolds [formula omitted]

We determine the characteristic rank of the canonical oriented vector bundle over G˜3,n for all n≥3, and as a consequence, we obtain the affirmative answer to a conjecture of Korbaš and Rusin. As an application of this result, we calculate the Z2-cup-length for a new infinite family of manifolds G˜3,n. This result confirms the corresponding claim of Fukaya's conjecture.

An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

In a previous work, the authors introduced the notion of `coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss-Bonnet formulas on coherent tangent bundles on 22-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal E$ be a vector bundle of rank $n$ over $M^n$, and $\varphi:TM^n\to \mathcal E$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss-Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\varphi$ under the assumption that $\varphi$ admits only certain kinds of generic singularities. We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

We prove that the Euler form of a metric connection on a real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$.

A note on $n$-gerbes and transgressions

In this note we provide an explicit interpretation of a class of (q-1) -gerbes with multi-layered connections in terms of transgression of a fibrewise closed q -form on a fibration to a closed (q+1) -form on the base manifold, with the basic example of the Euler class of an oriented vector bundle in mind ( q \geq 0 ). Picken's and Ferreira–Gothen’s n -gerbopoles are discussed from this point of view. Furthermore, string structures (à la Cocquereaux–Pilch and à la Spera–Wurzbacher) are briefly addressed and recast within the proposed framework.

Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

This Note describes sharp Milnor–Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern–Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to some other locally symmetric spaces they do admit interesting flat vector bundles in the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number. To cite this article: M. Bucher, T. Gelander, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Diffeomorphism of total spaces and equivalence of bundles

Let E 1 and E 2 be the total spaces of smooth, oriented vector bundles of rank k over the n-sphere. We show that if E 1 and E 2 are diffeomorphic, with orientation preserved, then the bundles are smoothly equivalent up to orientation of the base whenever k>[( n+1)/2]+1. With an additional hypothesis, the same conclusion holds when the base is an arbitrary closed, oriented n-manifold. Furthermore, if the base manifold is a homotopy n-sphere and if one of the bundles has a nowhere-zero cross-section, then the oriented bundles are smoothly equivalent up to orientation of the base in the case where k=[( n+1)/2]+1 as well. The latter statement is false if k<[( n+1)/2]+1, as several counterexamples illustrate. We show that each of these examples is an open manifold E admitting a complete metric of nonnegative sectional curvature for which the zero section of the nontrivial vector bundle, a standard sphere, is not the image of a soul in the sense of Cheeger and Gromoll under any diffeomorphism of E.

A Secondary Chern-Euler Class

Let xi be a smooth oriented vector bundle, with n-dimensional fibre, over a smooth manifold M. Denote by xi-hat the fibrewise one-point compactification of xi. The main purpose of this paper is to define geometrically a canonical element Upsilon(xi) in H^n(xi-hat,Q) (H^n(xi-hat,Z) tensor 1/2, to be more precise). The element \Upsilon(\xi) is a secondary characteristic class to the Euler class in the fashion of Chern-Simons.

Analytic and topological invariants associated to nowhere zero vector fields

It is well-known that the classical Gauss-Bonnet-Chern theorem [C] can be interpreted as an index theorem for de Rham-Hodge operators (cf. [BGV] and [LM]) and that it is nontrivial only for even dimensional manifolds. This paper arose from an attempt to search an odd dimensional analogue of this index theorem. Recall that the Gauss-Bonnet-Chern theorem is closely related to the famous Poincare-Hopf index formula expressing the Euler number as the sum of indices of singularities of a tangent vector field. Now for odd dimensional manifolds, or more generally for manifolds with vanishing Euler number, we take the advantage that another result of Hopf asserts that there always exist nowhere zero vector fields (see e.g. Steenrod [S]). Thus let M be an odd dimensional oriented closed manifold and let V be a nowhere zero vector field on M . Let γ be the one dimensional oriented vector bundle over M generated by V . Then TM/γ carries a canonically induced orientation. Let e(TM/γ) be the Euler class of TM/γ. For any integral element ω in H1(M,Q), we will take 〈ωe(TM/γ), [M ]〉 as our substitute for the Euler class appeared in the Gauss-Bonnet-Chern theorem. The main result of this paper gives an analytic formula for this number as the index of certain elliptic Toeplitz operators. In fact, a general odd index theory has already been developed by BaumDouglas [BD] who pointed out that the associated index can be computed

Intersections in hyperbolic manifolds

We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total spaces of orientable vector bundles over M.

On the classification of oriented vector bundles over 5-complexes

On the classification of oriented vector bundles over 5-complexes

On topological invariants of vector bundles

Let E →W be an oriented vector bundle, and let X(E) denote the Euler number of E. The paper shows how to calculate X(E) in terms of equations which describe E and W . Introduction. Let F = (F1, . . . , Fk) : R → R, n − k > 0, be a C-map such that W = F−1(0) is compact and rank[DF (x)] ≡ k at every x ∈ W . From the implicit function theorem W is an (n − k)-dimensional C-manifold. Let G1, . . . , Gs : R→R, where m = s+n−k, be a family of C-vector functions, and assume that the vectors G1(x), . . . , Gs(x) are linearly independent for every x ∈W . Define E = {(x, y) ∈W × R | y ⊥ Gi(x), i = 1, . . . , s} = { (x, y) ∈W × R ∣∣∣ ∑ yjGji (x) = 0, i = 1, . . . , s} . Clearly E is an (n− k)-dimensional vector bundle over W . In particular, if s = k and Gi = gradFi then E becomes TW . Later we shall describe how to orient W and E. Let X(E) be the Euler number of the bundle E (see [1], Chapter 5.2). The problem is how to calculate X(E) in terms of F and G1, . . . , Gs. Let SR = {(x, λ) ∈ R×R | ‖x‖ +‖λ‖ = R}, and let H : R×R → R × R be the map given by

On splitting topological R4-bundles

Let ξ 4 denote any topological R 4-bundle over a connected Poincaré complex X of formal dimension ⩽4. Let α be any bundle of lines or planes based on X. We determine necessary and sufficient conditions in terms of characteristic classes of ξ 4 and α, whenever possible, for ξ to split off a subbundle isomorphic to α. That is, the existence of a Whitney sum decomposition, ξ 4≈α⊤β where both α and β have positive fiber dimension, is stablished in terms of characteristic classes. In particular, our results extend the work of Hirzebruch and Hopf [3] on oriented vector bundles of dimension 4 to arbitrary R 4-bundles.

The classification of orientable vector bundles over CW-complexes of small dimension

SynopsisThe classification of orientable vector bundles over CW-complexes of dimension ≦8 is given in terms of characteristic classes using elementary homotopy theoretic methods and relations among characteristic classes.

An invariant for almost-closed manifolds

1. Let M be a compact, oriented, connected, ^-dimensional differential manifold with dM (boundary M) homeomorphic to the n—1 sphere Then dM represents an element [dM] of r~ , the group of differential structures (up to equivalence) on 5~. We consider the (much studied) problem of expressing [dM] in terms of computable invariants of M. Let 7Tn-i be the n — 1 stem, Jo: 7r»(BSO)—»7rw-i the classical J-homomorphism, and ir^x the cokernel of Jb. In [S], a map P: T-***-* was defined (see below). We will define an invariant A(M) which is a subset of w'n„x (and often consists of a single element). The main theorem states: P[dM]EA(M). In a strong sense, the definition of A(M) involves only homotopy theory. Moreover, A(M) seems amenable to computation by standard techniques of algebraic topology. We illustrate this below and, as applications, give explicit examples (1) of a manifold M, n odd, with [dilf]^0, and (2) of Mf n even, with [dM] not only 5*0, but in fact with [dM] not even contained in T^(dw)t the subgroup in r*** 1 of elements which bound 7r-manifolds. (Examples of M, n even, with [dM]^0 are of course well known.) Other applications, and detailed proofs, will appear elsewhere. REMARK 1. By [5], kernel P^T^idr). If n is odd, T^dr)**!), so P is injective, while if n^2 (4), kernel PQZ2. If wsO (4), kernel P tends to be large (but see §5). Let BSO, BSPL, BSTop be the stable classifying spaces for orientable vector bundles» piecewise-linear ( = PL) bundles, topological bundles. There are maps Jo: 7rn(BSG)—>7r»-i (G = 0, PL, Top) and a commutative diagram with exact rows

On trivialities of Euler classes of oriented vector bundles over manifolds

On trivialities of Euler classes of oriented vector bundles over manifolds