A Pamphlet on Motivic Cohomology

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Cohomology of the Steenrod algebra

Translated from Matematicheskie Zametki, Vol. 52, No. 2, pp. 120–126, August, 1992.     

The Cohomology of the Universal Steenrod Algebra

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.     

Cohomology of unipotent algebraic and finite groups and the Steenrod algebra

Dedicated to Professor T. Nakamura on his sixtieth birthday     

Classes de Chern des ensembles analytiques

Let V be a compact complex analytical subset of a holomorphic manifold M. We shall define classes in homology, which coincide, when V is non-singular, with the Poincaré duals $\text{c}{}^1\text{(N}{}_{\mathrm{V}}\text{)?[V]}$ and $\text{c}{}^1\text{(TV)?[V]}$ of the Chern classes of the normal bundle N_V and of the tangent bundle TV. However, these definitions depend in general on the data on a desingularization $\text{?}\text{:V′→V}$ of V, except in some particular cases, as complex curves or sets which are locally complete intersection (LCI). These classes make possible to generalize some theories already known for LCI, such as the various indices of foliations relatively to invariant subsets, or the Minor numbers and classes. To cite this article: V. Cavalier et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

模p Steenrod代数上同调的一些注记

模p Steenrod代数A的上同调H~(s,t)(A)是决定球面稳定同伦群的最有力数据.首先给出了模p Steenrod代数A和May谱序列的一些重要结论,而后给出与乘积元γ_(s+3)l_ng_0∈H~(s+8,t(s,n))(A)密切相关的May谱E_1项的结果,这些结论对该乘积元的非平凡性研究有重要意义,其中t(s,n)=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q+s, 0≤sp-6, n≥4, p≥11, q=2(p-1).     

Classes de Chern Des Ensembles Analytiques

Let V be a compact complex analytic subset of a non-singular holomorphic manifold M. Assume that V has pure complex dimension n. Denote by V₀ its regular part, and by [V] its fundamental class in H_2n(V; ). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*). If V is a locally complete intersection (LCI), it is known that the normal bundle N_{V_0} in M to V₀ in M has a natural extension N_V to all of V, so that we can define its Chern classes c^(*)(N_V) in cohomology, as well as the Chern classes c_vir^(*) (V) of the virtual tangent bundle T_vir(V):=[TM|_V - N_V] in the K-theory K⁰(V). This has applications

On Steenrod Operations in Oriented and Special Unitary Cobordism Theories

Algebras of Steenrod operations in oriented and special unitary cobordisms are constructed. The representing spectra of SO- and SU-cobordisms are studied. It is shown that on the corresponding spectra, there exists the structure of a multiplicative family over a suitable bioperad. The connection with Steenrod operations proposed by T. tom Dieck is established. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 19, Topology and Noncommutative Geometry, 2004.     

Bounded Classes in the Cohomology of Manifolds

We discuss the image of a natural homomorphism from the bounded cohomology to the ordinary cohomology of a manifold. We give a necessary and sufficient condition for a Haken 3-manifold M to have the property that any class in the nth cohomology of M is bounded if n > 1.The first author was partially supported by JSPS.     

On Ginzburg's Bivariant Chern Classes, II

For a morphism whose target variety is nonsingular, the Chern–Schwartz–MacPherson class homomorphism followed by capping with the pullback of the Segre class of the target variety is called the Ginzburg–Chern class. In this paper, using the Verdier–Riemann–Roch for Chern Class, we show that the correspondence assigning to a bivariant constructible function on any morphism with nonsingular target variety the Ginzburg–Chern class of it is the unique Grothendieck transformation satisfying the 'normalization condition' that for morphisms to a point it becomes the Chern–Schwartz–MacPherson class homomorphism, except for that the bivariant homology pullback is considered only for a smooth morphism.     

Classes de Chern bivariantes et classes polaires relatives

W. Fulton and R. MacPherson conjectured the existence and the uniqueness of bivariant Chern classes. J.-P. Brasselet and C. Sabbah have given each one a construction and these constructions coincide. In this paper we give a relation between these bivariant Chern classes and the relative polar classes.     

Classes de Hirzebruch et classes de Chern motiviques

Let X be a complex algebraic variety. We define and study new theories of characteristic classes, defined on the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. One of them, $T_y$ is a homology class version of the motivic measure and generalizes the corresponding Hirzebruch characteristic. It unifies the Chern class transformation of Schwartz and MacPherson, the Todd class transformation of Baum–Fulton–MacPherson and the L-class transformation of Cappell–Shaneson. To cite this article: J.-P. Brasselet et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

On the Algebra of Operations for Hopf Cohomology

In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liuleviciusdefined Steenrod squaring operations Sq^k on the cohomology ringof any cocommutative Hopf algebra over Z/2. Later, J. P. Mayshowed that these operations satisfy Adem relations, interpretedso that Sq⁰ is not the unit but an independent operation. Thus,these Adem relations are homogeneous of length two in the generators.This paper is concerned with the bigraded algebra B that isgenerated by elements Sq^k and subject to Adem relations; itshows that the Cartan formula gives a well-defined coproducton B. Also, it is shown that B with both multiplication andcomultiplication should be considered neither a Hopf algebranor a bialgebra, but another kind of structure, for which thename ‘algebra with coproducts’ is proposed in thepaper. 2000 Mathematics Subject Classification 55S10 (primary),55T15 (secondary).     

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

We describe the equivariant Chow ring of the wonderful compactification X of a symmetric space of minimal rank, via restriction to the associated toric variety Y. Also, we show that the restrictions to Y of the tangent bundle T _X and its logarithmic analogue S _X decompose into a direct sum of line bundles. This yields closed formulas for the equivariant Chern classes of T _X and S _X , and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.     

Stability of Hereditary Graph Classes Under Closure Operations

If is a subclass of the class of claw‐free graphs, then is said to be stable if, for any , the local completion of G at any vertex is also in . If is a closure operation that turns a claw‐free graph into a line graph by a series of local completions and is stable, then for any . In this article, we study stability of hereditary classes of claw‐free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.     

Steenrod squares in Cotor

It is first shown that the comparison theorem in relative homological algebra is the only tool to be used for introducing Steenrod operations in Cotor as well as in Ext. Then, the interaction of the operations in both functors is discussed in full detail. Some basic properties of the operations are given in the last section of the paper.The authors would like to thank a referee for his suggestion to reorganize their paper in the present form.This work is supported in part by NSF research grant GP-9585.