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Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back \sigma ^\sharp, each 2-cocycle \phi gives us the 3-cocycle \sigma ^\sharp \phi. For oriented classical links in the 3-space, we explore relation between their quandle 2-cocycle invariants associated with \phi and their shadow 3-cocycle invariants associated with \sigma ^\sharp \phi. For oriented surface links in the 4-space, we explore how powerful their quandle 3-cocycle invariants associated with \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed. [ABSTRACT FROM AUTHOR]

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial n -cocycles for Alexander biquandles. For a biquandle X , its geometric realization B X is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of B X is finitely generated if the biquandle X is finite. [ABSTRACT FROM AUTHOR]

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]

A theory of (co)homologies related to set-theoretic n-simplex relations is constructed in analogy with the known quandle and Yang-Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalise Hietarinta's idea of "permutation-type" solutions to the quantum (or "tensor") n-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial π2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras. [ABSTRACT FROM AUTHOR]

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples. [ABSTRACT FROM AUTHOR]

The JLO character formula due to Jaffe -Lesniewski-Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys. 112 (1988) 75-88] assigns to each Fred-holm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer-Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math. 199 (2006) 29-87] at the level of entire cyclic cohomology. [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]

We prove that for not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of is isomorphic to , where P and Q are the weight and root lattices of , respectively. This implies that the group of autoequivalences of the tensor category of -modules is the semidirect product of and the automorphism group of the based root datum of . For q=1 we also obtain similar results for all compact connected separable groups. [ABSTRACT FROM AUTHOR]

We define families of invariants for elements of the mapping class group of Σ, a compact-orientable surface. For any characteristic subgroup , let J(H) denote the subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H, we associate a higher-order ρψ-invariant and a signature 2-cocycle σψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular, each ρψ is a quasimorphism and each σψ is a bounded 2-cocycle on J(H). In one of the simplest nontrivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ. [ABSTRACT FROM PUBLISHER]

We identify Čech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L ∞-algebra. Applications to deformation theory are described. [ABSTRACT FROM PUBLISHER]

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]

We develop a new KAM scheme that applies to SL(2, $${{\mathbb R}}$$) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters. [ABSTRACT FROM AUTHOR]

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to $H^2(\widehat{Z(G)};{\mathbb T})$. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to $H^2(\widehat{Z(G)};{\mathbb T})\rtimes {\rm Out}(G)$. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups. [ABSTRACT FROM AUTHOR]

We compute the third cohomology group of a dihedral quandle of odd order with finite field coefficient, generalizing our previous work. In particular, we obtain many new examples of non-trivial 3-cocycles. We check non-triviality of the associated invariant for knots. [ABSTRACT FROM AUTHOR]

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]

The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers. [ABSTRACT FROM AUTHOR]

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]

Let $\mathfrak{F}_{\lambda}^{n}$ be the $\mathop {\mathfrak {osp}}\nolimits \,(n|2)$-module of weighted densities on ℝ of weight λ. We compute the cohomology spaces $\mathrm{H}^{k}_{\mathrm{diff}}\left(\mathop {\mathfrak {osp}}\nolimits \,(n|2),\mathfrak{F}_{\lambda}^{n}\right)$, where k=1 and n=0,1,2 or k=2 and n=0,1. We explicitly give cocycles spanning these cohomology spaces. [ABSTRACT FROM AUTHOR]

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology. [ABSTRACT FROM AUTHOR]

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 consider a minimal, free action, φ, of the group ℤd on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both. [ABSTRACT FROM AUTHOR]

By using Stokes's theorem, we prove an Eichler cohomology theorem for generalized modular forms with some restrictions on the relevant multiplier systems. [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]

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions. [ABSTRACT FROM AUTHOR]

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]

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations. [ABSTRACT FROM AUTHOR]

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]

In this paper, we give a partial answer to the problem proposed by Lan Wen. Roughly speaking, we prove that for a fixed i, f has C 1 persistently no small angles if and only if f has a dominated splitting of index i on the C 1 i-preperiodic set P ( f). To prove this, we mainly use some important conceptions and techniques developed by Christian Bonatti. In the last section, we also give a characterization of the finest dominated splitting for linear cocycles. [ABSTRACT FROM AUTHOR]

Fox's shadow p-colorings of a knot K define two kinds of Laurent polynomials, Φp(K) and $\Phi_{p}^{*}(K) \in \mathbb{Z}[t^{\pm1}]/(t^p-1)$, as invariants of K. We prove that the equality $\Phi_p(K) = p\Phi_{p}^{*}(K)$ holds for any knot K. Also we prove that, if the p-coloring number of K is equal to p2, then $\Phi_{p}^{*}(K)$ has the form $p \sum_{i=0}^{p-1} t^{Ni^{2}}$ for some N ∈ ℤp. [ABSTRACT FROM AUTHOR]

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]

We establish conditions for aperiodicity of cocycles (in the sense of [12]), obtaining, via a study of perturbations of transfer operators, conditional local limit theorems and exactness of skew-products. Our results apply to a large class of Markov and non-Markov interval maps, including beta transformations. This allows us to establish various stochastic properties of beta expansions. [ABSTRACT FROM AUTHOR]

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]

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise. [ABSTRACT FROM AUTHOR]

We introduce Virasoro‐type extensions for the Neretin spheromorphism groups Np (for each integer p ≥ 2), the combinatorial or p‐adic (when p is prime) analogues of the diffeomorphism group of the circle. The proof of their non‐triviality is based on the computation of the Schur multiplier H2 (Vp, Z) of the Higman–Thompson groups Vp, which are countable subgroups of Np. [ABSTRACT FROM PUBLISHER]

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]