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Abstract We give a uniform construction of the higher indices of elliptic operators associated to Alexander–Spanier cocycles of either parity in terms of a pairing à la Connes between the K -theory and the cyclic cohomology of the algebra of complete symbols of pseudodifferential operators, implemented by means of a relative form of the Chern character in cyclic homology. While the formula for the lowest index of an elliptic operator D on a closed manifold M (which coincides with its Fredholm index) reproduces the Atiyah–Singer index theorem, our formula for the highest index of D (associated to a volume cocycle) yields an extension to arbitrary manifolds of any dimension of the Helton–Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres. In fact, the totality of higher analytic indices for an elliptic operator D amount to a representation of the Connes–Chern character of the K -homology cycle determined by D in terms of expressions which extrapolate the Helton–Howe formula below the dimension of M. [ABSTRACT FROM AUTHOR]

We introduce a modified homology and cohomology theory for involutory biquandles (also known as bikei). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in R4. [ABSTRACT FROM AUTHOR]

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed. [ABSTRACT FROM AUTHOR]

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki from their 2009 paper. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Finally, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle. [ABSTRACT FROM AUTHOR]

Abstract: We study existence, uniqueness and triviality of path cocycles in the quantum Cayley graph of universal discrete quantum groups. In the orthogonal case we find that the unique path cocycle is trivial, in contrast with the case of free groups where it is proper. In the unitary case it is neither bounded nor proper. From this geometrical result we deduce the vanishing of the first -Betti number of . [Copyright &y& Elsevier]

Abstract: Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns ‘quantities’ to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given. [Copyright &y& Elsevier]

Abstract: We prove that for not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of and is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence. [Copyright &y& Elsevier]

Abstract: We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids. [Copyright &y& Elsevier]

Abstract: Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the -algebras and of the -theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these -algebras are traceless. Nevertheless, harmonic currents give rise to -cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling. [Copyright &y& Elsevier]

This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus Tn = Rn/Zn and H is a real-analytically completely integrable convex hamiltonian on T*Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a real-analytically completely integrable riemannian metric hamiltonian. [ABSTRACT FROM AUTHOR]

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links. [ABSTRACT FROM AUTHOR]

Abstract: We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid G-extensions, which we call “connections on gerbes”, and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for G-gerbes over stacks. We also introduce G-central extensions of groupoids, generalizing the standard groupoid -central extensions. As an example, we apply our theory to study the differential geometry of G-gerbes over a manifold. [Copyright &y& Elsevier]

Abstract: We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation – so called -derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants – we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction. [Copyright &y& Elsevier]

Abstract: We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We apply this study to give a new purely algebraic construction of the Casson invariant. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group. [Copyright &y& Elsevier]

Abstract: Using Barnes–Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape . The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters , a Nordstrom–Robinson code, and gives a context for determining its automorphism group, which has form . [Copyright &y& Elsevier]

Abstract: We give a cocycle expansivity result for multimodal interval maps with non-flat critical points. It extends the Mañé hyperbolicity theorem to also describe orbits which pass near critical points. To cite this article: N. Dobbs, C. R. Acad. Sci. Paris, Ser. I 345 (2007). [Copyright &y& Elsevier]

We introduce a new type of generalized derivations associate with Hochschild 2-cocycles and prove that every generalized Jordan derivation of this type is a generalized derivation under certain conditions. This result contains the results of I. N. Herstein [6, Theorem 3.1] and M. Ashraf and N-U. Rehman [1, Theorem]. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish the pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers–Ulam stability of some functional equations. We prove that for each Banach algebra A, Banach A-bimodule X and positive integer if and only if the nth cohomology group approximately vanishes. [Copyright &y& Elsevier]

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W t − I ∈ S 2 (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations. [ABSTRACT FROM AUTHOR]

Abstract: Using the language of stacks one can give a simple definition of functorial Chern classes for twisted sheaves. Calculating the cohomology ring of a -gerbe we observe that the twisted Chern classes used by Huybrechts and Stellari are specializations of these classes. We describe explicitly the relation between the choice of a cocycle in the definition of twisted sheaves and the 2-categorical structure of -gerbes. To cite this article: J. Heinloth, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

A new approach to the computation of the cohomology of groups acting on Buekenhout geometries is described. It is shown that in many cases, including BN-pairs and groups acting on dimensional linear spaces, to compute the kth cohomology group Hk(G,M) it is sufficient, roughly speaking, to consider only parabolic subgroups of rank ⩽k+1. As an application it is shown that Hk(An,Fp)=0 for k⩾1, p>k+2 (where An is the alternating group). [Copyright &y& Elsevier]

We give explicit polynomial generators for the homology rings of BSU and BSpin for complex oriented theories. Using these we are able to provide an alternative proof of the result of Hopkins et al. for symmetric 2-cocycles on formal group laws. [Copyright &y& Elsevier]

If we have a flow $(X,\Bbb{Z}^m)$ and a cocycle $h$ on this flow, $h:X\times \Bbb{Z}^m\rightarrow \Bbb{R}^m$, then $h$ is called {\it close to linear} if $h$ can be written as the direct sum of a linear (constant) cocycle and a cocycle in the closure of the coboundaries. Many of the desirable consequences of linearity hold for such cocycles and, in fact, a close to linear cocycle is cohomologous to a cocycle which is norm close to a linear one. Furthermore in the uniquely ergodic case all cocycles are close to linear. We also establish that a close to linear cocycle which is covering is cohomologous to one with the special property that it can be extended by piecewise linearity to an invertible cocycle from $X\times \Bbb{R}^m$ to itself. This implies that a suspension obtained from a close to linear cocycle is isomorphic to a time change of the suspension obtained from the identity cocycle. [ABSTRACT FROM AUTHOR]