Secondary Characteristic Classes Research Articles

Flat bundles and residues forfoliations

In this note we compute the residues for a vector field preserving a foliation. We assume the vector field disappears only to first order along its singular set and that its singular set consists of closed separated leaves. Under these conditions, the residue is completely determined by the characteristic classes of the flat bundle v o, which is the normal bundle v of the foliation restricted to the singular set, and the action of the vector field on v o. For a foliation of even codimension q whose normal bundle has Euler class zero, we interpret the Pontrjagin ring of the normal bundle in dimension 2q as an obstruction to the existence of such a vector field. We show that if the Euler class of the normal bundle of a foliation is zero and it admits such a vector field, then many of its secondary characteristic classes must be also be zero. This is one of the few results relating the geometry of a foliation to its characteristic classes. We will make extensive use of the material in [6]. It will be very helpful to the reader to be familiar with this paper.

The secondary characteristic classes of solvable foliations

The secondary characteristic classes of a codimension q q foliation of a manifold M M are certain cohomology classes of M M which are constructed from the curvature matrix of a torsion-free connection on the normal bundle of the foliation. We consider the foliation of a real or complex Lie group G G by the left cosets of a closed connected subgroup H H and any foliation obtained from it through dividing by a discrete subgroup Γ ⊂ G \Gamma \subset G . Such foliations are homogeneous. The Fuks-Pittie conjecture for homogeneous foliations is that the secondary classes are generated by { h 1 h i 2 ⋯ h i λ τ 1 q } \{ {h_1}{h_{{i_2}}} \cdots {h_{{i_\lambda }}}\tau _1^q\} . We prove the Fuks-Pittie conjecture if H H is reductive or solvable. We also prove an Addition Theorem for the classes which can be applied to reduce the general problem of calculating the secondary classes to the case in which G G is semisimple.

The secondary characteristic classes of parabolic foliations

The secondary characteristic classes of parabolic foliations

On the topological charge concentrated at a point

For SU(2) gauge fields over the 4-dimensional sphere with a finite number of points x 1, x 2, ..., and x N removed, there are gauge transformations which modify the topological charge concentrated at x j by adding n j , where n 1, n 2, …, and n N . are integers such that Σ N j = 1 n j = 0. However, the reduction modulo Z of the topological charge at a point is well defined, being given in terms of the secondary characteristic classes of Chern and Simons, except when the topological charge is indeterminate.

On nontrivial characteristic classes of contact foliations

In this article we give some nontrivial realizations of secondary characteristic classes of contact foliations.

Independent Variation of Secondary Classes

teristic classes of a certain class of foliations. The examples we compute show that many of these secondary classes vary linearly independently. This generalizes a result due to Thurston [T] on the variation of one of these classes, the Godbillon-Vey class. It is also a partial analogue of the results of Bott [B2] and Lazarov-Pasternack [LP] on the independent variation of the secondary classes for holomorphic and Riemannian foliations respectively. We are also able to compute the Simons' characters of these foliations and we show that many of them vary linearly independently. The method we use is a generalization of the theory of residues of singular foliations due to Baum-Bott [BB]. We work with the natural foliation z on a flat vector bundle and a vector field X tangent to the fiber of the bundle which preserves the foliation. We assume the vector field has an isolated singularity along the zero section. This situation determines certain cohomology classes on the zero section, the residues of z- and X. We relate these residues to the secondary characteristic classes and Simons' characters of the foliation ' spanned by z and X off the zero section. Using this technique we compute examples which show that the residues of Z and X and thus the characteristic classes of Z vary linearly independently. In Section 2 we record some facts we will use later. In Section 3 we

Derivatives of secondary characteristic classes

Derivatives of secondary characteristic classes

Residues and secondary characteristic classes for projective foliations

Residues and secondary characteristic classes for projective foliations

A remark on the continuous variation of secondary characteristic classes for foliations

A remark on the continuous variation of secondary characteristic classes for foliations

Secondary characteristic classes for Riemannian foliations

Secondary characteristic classes for Riemannian foliations

Characteristic classes for the deformation of flat connections

In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection ω 0 {\omega _0} . If f is a diffeomorphism of M, then ω 1 = f ∗ ω 0 {\omega _1} = {f^\ast }{\omega _0} is another flat connection. Denote by α \alpha the difference of these two connections. Then α \alpha and its exterior covariant derivative D α D\alpha are both tensorial forms on M. To each invariant polynomial φ \varphi of GL ( n , R ) {\text {GL}}(n,{\text {R}}) , where n = dim ⁡ M , φ ( α ; D α ) n = \dim M,\varphi (\alpha ;D\alpha ) is a globally defined form on M. The class { φ ( α ; D α ) } ∈ H ( M ; R ) \{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text {R}}) for deg ⁡ φ > 1 \deg \varphi > 1 gives rise to an obstruction of the deformability from ω 0 {\omega _0} to ω 1 {\omega _1} . In particular, we prove that ( + ) ( + ) and ( − ) ( - ) connections, in the sense of E. Cartan, cannot be deformed to each other.

Deformations of secondary characteristic classes

IN THIS paper we exhibit a formula for the derivative of a deformation of a secondary characteristic class. These classes were originally defined by Godbillon and Vey [7] and Chern and Simons [6] and have been the subject of great interest in the theory of foliations. See, for example, Bott and Haefliger [4] and Simons [IO]. The formula implies that certain of the generalized Godbillon and Vey characteristic classes for foliations are invariants of the connected components of the space of foliations on a manifold. In [12], D. Lehmann gives a proof of this theorem without using the derivative formula. We remark that the cohomology classes of a codimension q foliation coming from the Simons characters [IO] are also constant on connected components unless the character occurs in dimension 2q + 1. For real foliations, the examples of Thurston [I l] show that in dimension 2q + 1 it is possible to have classes which vary continuously. For examples in the complex case see Bott [2]. The formula is a generalization of a formula of Chern and Simons [6]. However, their formula was for forms defined on the principal bundle over a manifold, while ours is for forms defined on the manifold itself and so can be applied to the theory of secondary classes. The reader should also note that while the results here are stated only for BT, and WO,, similar results hold for the spaces associated to complex foliations and foliations with trivial normal bundles.

On secondary characteristic classes in cobordism theory

On secondary characteristic classes in cobordism theory

Secondary Characteristic Classes in K-Theory

Secondary Characteristic Classes in K-Theory

Secondary characteristic classes in $K$-theory

The object of this paper is to develop a general theory of secondary characteristic classes and to study secondary characteristic classes that arise in K-theory. Secondary characteristic classes are particularly adapted to studying embedding problems. Massey, and Peterson and Stein developed and exploited secondary characteristic classes in ordinary cohomology theory [11], [17], [18]. On the other hand, Atiyah [3] showed that certain primary characteristic classes yi in K-theory give good results for nonembedding of projective spaces. The characteristic classes we develop here were motivated by the desire to define a secondary operation when the top yi-class vanishes, in analogy with the operations which arise from the relation Sqi(wn) = wi u wn where wi is the ith Whitney class of an n-plane bundle. The viewpoint we will take in this paper is that, in a general cohomology theory, secondary characteristic classes arise in two ways: from a relation between characteristic classes and cohomology operations, or from the degeneration of the Gysin sequence. The organization of the paper is as follows. The first section is preliminary and collects results on spectra, functional operations, representation theory, and K-theory. In ?2 we give the various definitions of secondary characteristic classes in the setting of general cohomology theories and principal G-bundles. In ?3 we develop the crucial Peterson-Stein formula relating a functional operation in the universal example and the universal secondary characteristic class. We apply this formula in ?4 and ?5 to study the indeterminacy for the universal secondary characteristic classes and to study the relationship between the various definitions given in ?2. In ?6 we discuss the secondary characteristic classes in K-theory which arise from the relation /k(A _) =k -1, and in ?7 and ?8 we carry out some computations involving these operations. We should note that throughout this paper H* will always be a generalized cohomology theory. We hope that a forthcoming paper will deal with the application of these operations to embedding problems. Most of the material in this paper appeared in the author's doctoral thesis at Harvard University written under the direction of Professor Raoul Bott. The author wishes to express his gratitude to Professor Bott and also to Professors Donald Anderson and Al Vasquez.

Secondary Characteristic Classes for k-Special Bundles

Secondary Characteristic Classes for k-Special Bundles

Secondary characteristic classes for 𝑘-special bundles

1. Introduction. The secondary characteristic classes of a sphere bundle with vanishing Euler class were introduced by Massey [10]. These classes provide the additional information over and above the cohomology structure of the base needed to determine the mod 2 cohomology of the total space as ring, as module over the cohomology of the base, and as module over the mod 2 Steenrod algebra. It was for this purpose that these classes were originally defined. Subsequent work by Peterson and Stein [16], Meyer [13], and the author [14] led to many new properties of secondary characteristic classes, including analogues of the Wu and Whitney formulas and reinterpretation of these classes in terms of functional cohomology operations (yielding in turn simpler proofs of formulas of [16]). In this paper we define secondary characteristic classes for an n-plane bundle e such that the top k Whitney classes Wn-kk 1,. . ., wn of e vanish. For k = 1, these classes are the classes of Massey. By a theorem of Massey and Peterson [12], these classes are the additional information needed to determine the cohomology of the total space of the associated bundle with fibre Vn,k We then prove analogues of the Wu and Whitney formulas. Our secondary classes will be defined in terms of a functional cohomology operation, involving relative cup-product, defined by the basic method of Steenrod [17] on Thom classes of the bundle. While our formal treatment of the algebraic properties of such operations, as well as the presentation of the Wu and Whitney formulas, has been greatly influenced by the paper of Meyer [13], the operations we consider have smaller indeterminacy and the results obtained from these operations

Secondary Characteristic Classes

Secondary Characteristic Classes

Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction

What are called secondary characteristic classes in Chern–Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth $\infty$-groups, i.e., by smooth groupal $A_\infty$- spaces. Namely, we realize differential characteristic classes as morphisms from $\infty$-groupoids of smooth principal $\infty$-bundles with connections to $\infty$-groupoids of higher $U(1)$-gerbes with connections. This allows us to study the homotopy fibres of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.