We decompose θ(M), the twisted index obstruction to a positive scalar curvature metric for closed oriented manifolds with spin universal cover, into a pairing of a twisted K-homology with a twisted K-theory class and prove that θ(M) does not vanish if M is a closed orientable enlargeable manifold with spin universal cover.

We study the coniveau spectral sequence for quadrics defined by Pfister forms. In particular, we explicitly compute the motivic cohomology of anisotropic quadrics over ℝ, by showing that their coniveau spectral sequences collapse from the -term

This article gives a complete classification of generically split projective homogeneous varieties. This project was begun in our previous article [PS10], but here we remove all restrictions on the characteristic of the base field, give a new uniform proof that works in all cases and in particular includes the case PGO2n+ which was missing in [PS10].

Bases are constructed for the homogeneous components of the invariant subalgebra and anti-invariant submodule of the algebra K*K(p) of cooperations for p-local complex K-theory.

For a field k of cohomological dimension d we prove that the groups , (l, car.k) = 1, are birational invariants of smooth projective geometrically integral varieties over k of dimension n. Using the Kato conjecture, which has been recently established by Kerz and Saito [18], we obtain a similar result over a finite field for the groups . We relate one of these invariants with the cokernel of the l-adic cycle class map , which gives an analogue of a result of Colliot-Thélène and Voisin [5] 3.11 over ℂ for varieties over a finite field.

Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.

We introduce cohomologically triangulated categories as triples (A,t,▽) given by an additive category A, an additive equivalence t:AA and a cohomology class ▽ in the translation cohomology H3(A,t). A stable homotopy theory C with A = HoC yields such a triple and the class of distinguished triangles in A is deduced from ▽.

In this paper we introduce new local symbols, which we call 4-function local symbols. We formulate reciprocity laws for them. These reciprocity laws are proven using a new method - multidimensional iterated integrals. Besides providing reciprocity laws for the new 4-function local symbols, the same method works for proving reciprocity laws for the Parshin symbol. Both the new 4-function local symbols and the Parshin symbol can be expressed as a finite product of newly defined bi-local symbols, each of which satisfies a reciprocity law. The K-theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and the K-theoretic variant of the 4-function local symbol satisfy reciprocity laws, whose proof is based on Milnor K-theory (see the Appendix). The relation of the 4-function local symbols to the double free loop space of the surface is given by iterated integrals over membranes.

Let k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZXsf which is stable under pullback and that all the slices have a canonical structure of strict modules over HZXsf. If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZX ⊗ ℚ and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.

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.

Let X be a reduced and projective singular surface over ℂ and let → X be a resolution of singularities of X. We show that CH2(X) ≅ CH2() if and only if for i = 0, 1. This verifies a conjecture of Srinivas.

We also verify Bloch's conjecture for singular surfaces assuming it holds for smooth surfaces. As a byproduct, we give an application to projective modules on certain singular affine surfaces.

It is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.

Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.

In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.

The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].

We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an anti-automorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(ℤ[½]).

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.

Given a suitable functor T : → between model categories, we define a long exact sequence relating the homotopy groups of any X ε with those of TX, and use this to describe an obstruction theory for lifting an object G ε to . Examples include finding spaces with given homology or homotopy groups.

In [HL99], the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then shown that the associated heat operator converges to the Chern character of the index bundle of the operator. In [BH08], this result was improved by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.

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 show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.