Characteristic classes and transfer relations in cobordism

Decompositions of products of the Ray elements by free generators of small dimensions in the symplectic cobordism ring are obtained. In particular it is stated that most of the -dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.

     

On uniqueness of characteristic classes

We give an axiomatic characterization of maps from algebraic K-theory. The results apply to a large class of maps from algebraic K-theory to any suitable cohomology theory or to algebraic K-theory. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Adams operations and λ-operations on higher algebraic K-theory. We show that the Adams operations and λ-operations defined by Grayson agree with the ones defined by Gillet and Soulé.     

On tertiary exotic characteristic classes

We prove the existence and nontriviality of tertiary exotic characteristic classes extending the results of Peterson and Ravenel for secondary exotic classes.

     

On characteristic classes of Q-manifolds

Characteristic classes are defined for supermanifolds equipped with a homological vector field Q. We construct an infinite series of characteristic classes defined in terms of the second covariant derivatives of Q.     

Realizing characteristic classes by manifolds

Let BG be the classifying space for stable spherical fibrations, and let V be a finite dimensional vector subspace of the cohomology algebra H¹(BG; $\text{Z}$₂). We prove that V may be realized by a Poincaré duality space P, which means that if v:P→BG is the Spivak fibration, then v¹ maps V isomorphically onto its image in H¹(P;$\text{Z}$₂). By construction, P is the product of a certain Grassman manifold and a spherical fiber space over a closed smooth manifold.     

On exotic characteristic classes for spherical fibrations

We define for every so-called admissible relation r in the Steenrod algebra $\text{A}$ and for every oriented spherical fibration ξ over a CW-space an exotic characteristic class (mod 2) ε(r)(ξ), which is primitive and vanishes for sphere bundles. The set of exotic classes associated with the universal spherical fibration and the admissible Adem relations are compared with the algebra generators of H¹(BSG;$\text{Z}$₂) due to Milgram. Moreover, their behaviour under the action of $\text{A}$ is computed. Finally, we give a secondary Wu formula for exotic classes of special Poincaré duality spaces.     

Cobordism characteristic classes

Sunto. Viene generalizzata al caso di varietà PL la nota costruzione dei cicli rappresentativi delle classi caratteristiche di una varietà differenziabile come autointersezioni del fibrato tangente proiettificato. Tale generalizzazione che conduce a classi caratteristiche valide sia in coomologia che in cobordismo, si basa su una costruzione geometrica dei quadrati di Steenrod.

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Entrata in Redazione il 19 aprile 1977.

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Research made when the author was a member of the G.N.S.A.G.A. of the C.N.R.     

Characteristic numbers and cobordism classes of fiberings with fiber

Some groups of -dimensional unoriented cobordism classes represented by the total space of a fibering with the real projective space as fiber are determined.

     

On cobordism of manifolds with corners

This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.

     

On the divisibility of characteristic classes of non-oriented surface bundles

In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of non-oriented surface bundles are not divisible in the stable range.     

On the divisibility of characteristic classes of non-oriented surface bundles

In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of non-oriented surface bundles are not divisible in the stable range.     

Singular fibers and characteristic classes

Let be a proper generic map between smooth manifolds with dimN−dimM=−1. We explicitly calculate the cohomology class dual to the closure of the set of points in N over which lies a specific singular fiber in terms of characteristic classes of M and N.     

Secondary characteristic classes of super-foliations

We construct secondary classes for super-foliations of codimension $0+\mathrm{\epsilon }1$ and $1+\mathrm{\epsilon }1$. We indicate how to generalize this construction for any regular super-foliations on super-manifolds. We interpret the secondary classes as classes of foliated flat connections. To cite this article: C. Laurent-Gengoux, C. R. Acad. Sci. Paris, Ser. I 339 (2004).