Cohomological Invariants Research Articles

Gribov horizon and spontaneous BRST symmetry breaking

An equivalent formulation of the Gribov--Zwanziger theory accounting for the gauge fixing ambiguity in the Landau gauge is presented. The resulting action is constrained by a Slavnov-Taylor identity stemming from a nilpotent exact Becchi-Rouet-Stora-Tyutin (BRST) invariance which is spontaneously broken due to the presence of the Gribov horizon. This spontaneous symmetry breaking can be described in a purely algebraic way through the introduction of a pair of auxiliary fields which give rise to a set of linearly broken Ward identities. The Goldstone sector turns out to be decoupled. The underlying exact nilpotent BRST invariance allows us to employ BRST cohomology tools within the Gribov horizon to identify renormalizable extensions of gauge invariant operators. Using a simple toy model and appropriate Dirac bracket quantization, we discuss the time evolution invariance of the operator cohomology. We further comment on the unitarity issue in a confining theory, and stress that BRST cohomology alone is not sufficient to ensure unitarity, a fact that, although well known, is frequently ignored.

Cohomological invariants of central simple algebras of degree 4

In this paper, we prove a result of Rost, which describes the cohomological invariants of central simple algebras of degree 4 with values in μ2 when the base field contains a square root of −1.

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

The C*-algebra of a vector bundle

We prove that the Cuntz–Pimsner algebra OE of a vector bundle E of rank ≧ 2 over a compact metrizable space X is determined up to an isomorphism of C(X)-algebras by the ideal (1 − [E])K0(X) of the K-theory ring K0(X). Moreover, if E and F are vector bundles of rank ≧ 2, then a unital embedding of C(X)-algebras OE ⊂ OF exists if and only if 1 − [E] is divisible by 1 − [F] in the ring K0(X). We introduce related but more computable K-theory and cohomology invariants for OE and study their completeness. As an application we classify the unital separable continuous fields with fibers isomorphic to the Cuntz algebra Om + 1 over a finite connected CW complex X of dimension d ≦ 2m + 3 provided that the cohomology of X has no m-torsion.

Dehn twists and invariant classes

A degeneration of compact Kaehler manifolds gives rise to a monodromy action on Betti moduli space H^1(X, G) = Hom(\pi_1(X),G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes that do not arise from global ones for reductive G. These examples exist even when G is abelian, as long as G contains multiple torsion points. Finally, for general G, we make a new conjecture on local invariant classes and produce some suggestive examples.

Coniveau spectral sequences of classifying spaces for exceptional and Spin groups

AbstractLetkbe an algebraically closed field ofch(k) = 0 andGbe a simple simply connected algebraic groupGoverk. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spacesBG.

Cohomological invariants for orthogonal involutions on degree 8 algebras

AbstractUsing 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.

THE COHOMOLOGICAL BIRATIONAL INVARIANTS FOR 4-DIMENSIONAL TORI

This paper is the first systematic step to obtain the birational classificationof 4-dimensional tori. All the cohomological birational invariants of this tori arecalculated. All the 4-dimensional tori with nontrivial invariants are described.

On the regular representation of an (essentially) finite 2-group

The regular representation of an essentially finite 2-group G in the 2-category 2Vect k of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it are computed. It is next shown that all hom-categories in Rep 2Vect k ( G ) are 2-vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ω : Rep 2Vect k ( G ) → 2Vect k is representable with the regular representation as representing object. As a consequence we obtain a k-linear equivalence between the 2-vector space V ect k G of functors from the underlying groupoid of G to V ect k , on the one hand, and the k-linear category E nd ( ω ) of pseudonatural endomorphisms of ω , on the other hand. We conclude that E nd ( ω ) is a 2-vector space, and we (partially) describe a basis of it.

An application of ample vector bundles in real algebraic geometry

Let E be an algebraic vector bundle on a compact nonsingular real algebraic set X, and let Z be the zero locus of a generic algebraic section of E. We investigate how certain cohomological invariants of X and Z are related. A crucial role in the proof is played by ample vector bundles on a suitable complexification of X.

On certain cohomological invariants of groups

Let R be a left and right ℵ 0 -Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants silp Z G and spli Z G , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if cd ̲ G = 0 , where cd ̲ is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].

Duality theorem for motives

A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological properties of the category of motives. The concept of a motive was introduced by Alexander Grothendieck in 1964 in order to formalize the notion of universal (co-)homology theory (see the detailed exposition of Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of motives DM−, constructed by Voevodsky [12] for algebraic varieties. The Poincare duality is a classical and fundamental result in algebraic topology that initially appeared in Poincare’s first topological memoir “Analysis Situs” [9] (as a part of the Betti numbers symmetry theorem proof). The proof of the general duality theorem for extraordinary cohomology theories apparently belongs to Adams [1]. Our purpose in this paper is to establish a general duality theorem for the category of motives. Essentially, we extend the main statement of [8] to this category. Many known results can easily be interpreted in these terms. In particular, we get a generalization of the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed to the authors by Andrěi Suslin in a private communication. Being inspired by his work and Dold–Puppe’s category approach [2] to the duality phenomenon in topology, we decided to present a short, simple, and self-contained proof of a similar result for the category of motives. Our result might be viewed as a purely abstract theorem and rewritten in the spirit of “abstract nonsense” as a statement about some category with a distinguished class of morphisms. Essentially, what is required for the proof is the existence of finite fiber products and the terminal object in the category of varieties, a small part of the tensor triangulated category structure for motives, and finally, the existence of transfers for the class of morphisms generated by graphs of a special type (of projective morphisms). However, rather, we preferred to formulate all statements for motives of algebraic varieties in order to clarify the geometric nature of the construction and make possible applications easier. This led, in particular, to the appearence of the second (co)homology index responsible for twist with the Tate object Z(1) (see Voevodsky [12]). The only exception is the classical Example 2. 2000 Mathematics Subject Classification. Primary 14F42.

Cohomological invariants of Jordan algebras with frames

In a previous paper the author determined all possible cohomological invariants of Aut ( J ) -torsors in Galois cohomology with mod 2 coefficients, where J is a split central simple Jordan algebra of odd degree n ⩾ 3 . In the present paper we extend these results to a larger class of groups G n 1 , … , n d r indexed by r = 0 , 1, 2 or 3, and n i positive integers that add to n ⩾ 3 such that n 1 is odd. We also give precise values for the essential dimension of these groups.

Cohomological invariants: exceptional groups and spin groups

This volume concerns invariants of G-torsors with values in mod p Galois cohomology - in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology - for various simple algebraic groups G and primes p. The author determines the invariants for the exceptional groups F4 mod 3, simply connected E6 mod 3, E7 mod 3, and E8 mod 5. He also determines the invariants of Spinn mod 2 for n </= 12 and constructs some invariants of Spin14. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of Spinn is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.

Discriminant of Symplectic Involutions

We define an invariant of torsors under adjoint linear algebraic groups of type C_n-equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H^3(F, mu_2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.

Morse–Novikov cohomology of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering, with the monodromy acting on this covering by holomorphic homotheties. We define three cohomology invariants, the Lee class, the Morse–Novikov class, and the Bott–Chern class, of an LCK-structure. These invariants play together the same role as the Kähler class in Kähler geometry. If these classes coincide for two LCK-structures, the difference between these structures can be expressed by a smooth potential, similar to the Kähler case. We show that the Morse–Novikov class and the Bott–Chern class of a Vaisman manifold vanish. Moreover, for any LCK-structure on a manifold, admitting a Vaisman structure, we prove that its Morse–Novikov class vanishes. We show that a compact LCK-manifold M with vanishing Bott–Chern class admits a holomorphic embedding into a Hopf manifold, if dim C M ⩾ 3 , a result which parallels the Kodaira embedding theorem.

Triple products and cohomological invariants for closed 3-manifolds

Triple products and cohomological invariants for closed 3-manifolds

Cohomological invariants of odd degree Jordan algebras

AbstractIn this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp2n) = n + 1 for n odd.

LOCAL COHOMOLOGY AND THE VARIATIONAL BICOMPLEX

The differential forms on the jet bundle J∞E of a bundle E → M over a compact n-manifold M of degree greater than n determine differential forms on the space Γ(E) of sections of E. The forms obtained in this way are called local forms on Γ(E), and its cohomology is called the local cohomology of Γ(E). More generally, if a group [Formula: see text] acts on E, we can define the local [Formula: see text]-invariant cohomology. The local cohomology is computed in terms of the cohomology of the jet bundle by means of the variational bicomplex theory. A similar result is obtained for the local [Formula: see text]-invariant cohomology. Using these results and the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gelfand–Fuchs cohomology introduced in [4], we construct non trivial local cohomology classes in the important cases of Riemannian metrics with the action of diffeomorphisms, and connections on a principal bundle with the action of automorphisms.

On a relation between certain cohomological invariants

Let G be a group, spli Z G the supremum of the projective lengths of the injective Z G -modules and silp Z G the supremum of the injective lengths of the projective Z G -modules. The invariants spli Z G and silp Z G were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] in connection with the existence of complete cohomological functors. If spli Z G is finite then silp Z G = spli Z G = findim Z G [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and cd ¯ Z G ≤ spli Z G ≤ cd ¯ Z G + 1 , where cd ¯ Z G is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that cd ¯ Z G = vcd G if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ , Arch. Math. 89 (1) (2007) 24–32] that if spli Z G is finite then G admits a finite dimensional model for E ¯ G , the classifying space for proper actions. We conjecture that spli Z G = cd ¯ Z G + 1 for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type FP ∞ .