We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulæ for the computation of the E3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

A fibre product construction is used to give a description of quantum line bundles over the generic Podlés spheres obtained by gluing two quantum discs along their boundaries. Representatives of the corresponding K0-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The K0-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a significant simplification of earlier index computations.

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on : we define a bimonad on as an endofunctor B which is a monad and a comonad with an entwining λ : BB → BB satisfying certain conditions. This λ is also employed to define the category of (mixed) B-bimodules. In the classical situation, an entwining λ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws τ : BB → BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining λ and lead to an extension of the theory of braided Hopf algebras.

An antipode is defined as a natural transformation S : B → B with special properties. For categories with limits or colimits and bimonads B preserving them, the existence of an antipode is equivalent to B inducing an equivalence between and the category of B-bimodules. This is a general form of the Fundamental Theorem of Hopf algebras.

Finally we observe a nice symmetry: If B is an endofunctor with a right adjoint R, then B is a (Hopf) bimonad if and only if R is a (Hopf) bimonad. Thus a k-vector space H is a Hopf algebra if and only if Homk(H,−) is a Hopf bimonad. This provides a rich source for Hopf monads not defined by tensor products and generalises the well-known fact that a finite dimensional k-vector space H is a Hopf algebra if and only if its dual H* = Homk(H,k) is a Hopf algebra. Moreover, we obtain that any set G is a group if and only if the functor Map(G,−) is a Hopf monad on the category of sets.

In a recent paper, Merkurjev showed that for a smooth proper variety X over a field k, the functor M* ↦ A0(X, M0) from cycle modules to abelian groups is corepresented by a cycle module constructed on the Chow group of 0-cycles of X. We show that if “proper” is relaxed, the result still holds by replacing the Chow group of 0-cycles by the 0-th Suslin homology group of X.

We describe a construction of an object associated to the fundamental group of ℙ1 − {0, 1, ∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products in the motivic cohomology of the ground field.

It is shown that the set of orbits of the action of the elementary symplectic group on all unimodular rows over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary general linear group. This result is used to improve injective stability for K1 of the symplectic group over non-singular affine algebras.

Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-groups, analogous to orthogonal and symplectic topological K-theory.

The proofs use in an essential way higher KSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitian K-groups in terms of higher algebraic K-groups.

We also relate periodicity to étale hermitian K-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

We show that the groups Kn(RG;ℤ/m) are Bott-periodic for n ≥ 1 whenever G is a finite group, m is prime to |G|, R is a ring of S-integers in a number field and 1/m ∊ R.

Let K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler ℤ-bilinear form associated to C, the Euler characteristic χC(M,N) of left C-comodules M,N, and the defect ∂C(M,N) ∊ ℤ associated to any computable Euler pair (M,N) of left C-comodules. We show that (lgthM, lgthN) = χC(M,N) + ∂C(M,N), for any computable Euler pair (M,N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect ∂C(M,N) is zero and (lgthM,lgthN) = χC(M,N), if the comodules M,N are finite-dimensional.

This paper displays model structures for the category of pro-objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro-simplicial sets, in which the cofibrations are monomorphisms and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques. There is, in particular, an n-type structure for pro-simplicial presheaves, and also a model structure in which the map from a pro-object to its Postnikov tower is formally inverted.

In his 1973 paper [4] Quillen proved a resolution theorem for the K-Theory of an exact category; his proof was homotopic in nature. By using the main result of Nenashev's paper [3], we are able to give an algebraic proof of Quillen's Resolution Theorem for K1 of an exact category. We view this as an advance towards the goal of giving an essentially algebraic subject an algebraic foundation.

Given a group G and a ring R, Loday [Lod, 1976] described an assembly map αG : hn(BG;L(R)) → Kn(RG) where L(R) is a spectrum with nth space K0(SnR) × BGL(SnR)+ for n ≥ 0 and Kn(RG) = πn(BGL(RG)+ × K0(RG)). Hambleton and Pederson, [HP, 2004], indicate a proof that this map is isomorphic to the map on homotopy groups from the assembly map . We will complete the proof of this.

We prove the following result announced by V. Voevodsky. If S is a finite dimensional noetherian scheme such that S = ∪αSpec(Rα) for countable rings Rα, then the stable motivic homotopy category over S satisfies Brown representability.

Using cyclotomic specializations of equivariant K-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if , where the ni's are powers of a fixed prime number p, then the rank of an exceptional object on X is congruent to ±1 modulo p.

In this paper, we show that rationally trivial torsors under split smooth linear algebraic groups induce fibre sequences in -homotopy theory. The results allow geometric proofs of stabilization results for unstable Karoubi-Villamayor K-theories and a description of the second -homotopy group of the projective line.