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The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified 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]

In this paper, we introduce cell-forms on 픐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 픐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 픐0,n, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a ℚ-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations. [ABSTRACT FROM PUBLISHER]

This paper is a companion to the Boardman-Wilson paper on the ring spectrum P(n). When the prime is 2, this spectrum is not commutative, which introduces several complications. Here, we supply the necessary details of the relevant Hopf algebroids and Hopf ring for this case. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to exhibit fine structure for polyhedral products Z ( K ; ( X _ , A _ ) ) and polyhedral smash products Z ˆ ( K ; ( X _ , A _ ) ) where moment-angle complexes are special cases in which ( X , A ) = ( D 2 , S 1 ) . There are three main parts to this paper. (1) One part gives a natural filtration of the polyhedral product together with properties of the resulting spectral sequence in Theorem 2.15 . Applications of this spectral sequence are given. (2) The second part gives a homological decomposition of Z ˆ ( K ; ( X _ , A _ ) ) in case ( X _ , A _ ) consists of CW pairs by using part 1. (3) Applications to the ring structure of Z ( K ; ( X _ , A _ ) ) are given for CW-pairs ( X , A ) satisfying suitable freeness conditions. [ABSTRACT FROM AUTHOR]

In this paper we determine the homology and the cohomology groups of two properly discontinuous groups of isometries of the hyperbolic plane having non-compact orbit spaces and the fundamental group of a graph of groups with a finite vertex groups and no trivial edges by extending Lyndon's partial free resolution for finitely presented groups. For the first two groups, we obtain partial extensions and the corresponding homology. We also compute the corresponding cohomology groups for one of these groups. Finally we obtain homology and cohomology in all dimensions for the last of the above mentioned groups by constructing a full resolution for this group. [ABSTRACT FROM AUTHOR]

We define the categorical cohomology of a k-graph Λ and show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C*-algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C*-algebra is isomorphic to a twisted groupoid C*-algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C*-algebras are nuclear and belong to the bootstrap class. [ABSTRACT FROM AUTHOR]

We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre's diagonalization for cubical cells. An implementation in C++ of the introduced algorithms is available at together with some examples. The paper is self-contained as much as possible, and is written at a very elementary level, so that basic knowledge of algebraic topology should be sufficient to follow it. [ABSTRACT FROM AUTHOR]

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup. [ABSTRACT FROM AUTHOR]

Let ( M, M, N) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M and M along their submanifold N to obtain M M. In this paper, we consider the bilinear and cubic forms of H*( M M, ℤ) when dim M M = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M M and M∐ M. [ABSTRACT FROM AUTHOR]

This work is devoted to the result on symplectic nilpotent Lie algebras classified in a paper by [Y. Khakimdjanov, M. Goze and A. Medina, Symplectic or contact structures on Lie groups, Differential Geom. Appl. 21 (2004) 41-54]. We focus the attention on the investigation of their cohomology groups by using algebra tools. We give also an overview of deformation to those algebras of dim. ≤ 4. [ABSTRACT FROM AUTHOR]

The support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order. [ABSTRACT FROM PUBLISHER]

Let p be an odd prime and A be the mod p Steenrod algebra. For computing the stable homotopy groups of spheres with the classical Adams spectral sequence, we must compute the E2-term of the Adams spectral sequence, Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.. In this paper we prove that in the cohomology of A, the product Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is nontrivial for n ≤ 5, and trivial for n = 3, 4,where δ͂s+4 is actually Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. a described by Wang and Zheng, p ≤ 11, 0 ≥ s ≥ p - 4 and t(s, n) = 2 (p-1)[(s+2)+(s+4)p+(s+3)p²+(s+4)p+p]. [ABSTRACT FROM AUTHOR]

Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the (co)homology of any commutative algebra with symmetric coefficient bimodules, however traditional Hochschild (co)homology is defined for any associative algebrawith not necessarily symmetric coefficient bimodules. In the present paper, we generalize higher orderHochschild (co)homology toworkwith associative algebraswhich need not be commutative and in particular, show that simplicial sets admit such a generalization if and only if they are one dimensional. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give the dimension and an algorithm to compute a basis of all the infinitesimal deformations of on the variety of -dimensional Lie algebra laws . Recall that every filiform Lie algebra can be obtained by a deformation of [Vergne (1970) ]. In the same way as filiform Lie algebras, all filiform Lie superalgebras can be obtained by infinitesimal deformations of the model Lie superalgebra . In this paper we will also study the infinitesimal deformations of which lie in , giving the dimension and an algorithm to compute a basis of them. One could think that the two sets of deformations aforementioned, one for Lie algebras and another for Lie superalgebras, can be the same. But this assumption is not correct, in particular we will prove that the set of deformations for Lie superalgebras is a strict subset of the set of deformations for Lie algebras. Thus, we will give a necessary and sufficient condition for a cocycle of the Lie algebra to be a cocycle of the Lie superalgebra . [Copyright &y& Elsevier]

Abstract: The main result of this paper is an application of the topology of the space to obtain results for the cohomology of the symmetric group on d letters, , with ‘twisted’ coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces with mod-p coefficients to describe the homology as a module for the general linear group over an algebraically closed field k of characteristic p. As a direct application, these results provide a method of reducing the computation of (where , are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many d, a complete determination of can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary d the authors determine for . An interesting phenomenon is uncovered-namely a stability result reminiscent of generic cohomology for algebraic groups. For each i the cohomology stabilizes as a increases. The methods in this paper are also powerful enough to determine, for any p and λ, precisely when . Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus. [Copyright &y& Elsevier]

Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol. 56 (part 1), pp. 1–26, AMS, Providence, ) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, ) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, ), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in (Duan and Zhao, and ) to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes. [ABSTRACT FROM AUTHOR]

Abstract: Our goal in this paper is to prove that, under appropriate hypotheses, the sum of the Betti numbers of a 1-connected elliptic space is greater or equal to the dimension of its vector space of homotopy. The paper concludes with some examples for which the inequality is strict. [Copyright &y& Elsevier]

Steenrod operations were defined by Voedvodsky in motivic cohomology in order to prove the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings [5]. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology. [ABSTRACT FROM AUTHOR]

Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes. [ABSTRACT FROM AUTHOR]

Let G be a semisimple Lie group with a maximal torus T. We present an explicit formula for the Borel transgression τ: H¹(T) → H²(G=T) of the fibration G → G=T. This formula corrects an error in the paper [9], and has been applied to construct the integral cohomology rings of compact Lie groups in the sequel works [4, 6]. [ABSTRACT FROM AUTHOR]

Let L and A be Lie triple systems, and let θ A be a representation of L on A. We first construct the third-order cohomology classes by derivations of A and L , then obtain a Lie algebra G θ A with a representation Φ on H 3 (L , A) , where θ A is given by an abelian extension 0 ⟶ A ⟶ L ~ → π L ⟶ 0. We study obstruction classes for extensibility of derivations of A and L to those of L ~. An application of Φ is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra . We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators . In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to , showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even (). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for . A similar strategy is used for the tensor algebra of a Banach space. [Copyright &y& Elsevier]

Abstract: In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in Boe et al. (2010) and Bagci et al. (2008) , about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to provide cohomologies of -ary Hom–Nambu–Lie algebras governing central extensions and one parameter formal deformations. We generalize to -ary algebras the notions of derivation and representation introduced by Sheng for Hom–Lie algebras. Also we show that a cohomology of -ary Hom–Nambu–Lie algebras could be derived from the cohomology of Hom–Leibniz algebras. [Copyright &y& Elsevier]

Abstract: In this paper we study the infinitesimal deformations of the -color Lie superalgebra . By means of these deformations all filiform -color Lie superalgebras can be obtained. In particular, we give a method that will allow us to determine the dimension of the subspaces that are composed by linearly integrable deformations. [Copyright &y& Elsevier]

Abstract: Additive deformations of bialgebras in the sense of Wirth are deformations of the multiplication map of the bialgebra fulfilling a compatibility condition with the coalgebra structure and a continuity condition. Two problems concerning additive deformations are considered. With a deformation theory a cohomology theory should be developed. Here a variant of the Hochschild cohomology is used. The main result in the first part of this paper is the characterization of the trivial deformations, i.e. deformations generated by a coboundary. Starting with a Hopf algebra, one would expect the deformed multiplications to have some analogue of the antipode, which we call deformed antipodes. We prove, that deformed antipodes always exist, explore their properties, give a formula to calculate them given the deformation and the antipode of the original Hopf algebra and show in the cocommutative case, that each deformation splits into a trivial part and into a part with constant antipodes. [Copyright &y& Elsevier]

Abstract: We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold X. Then we show that the stack of objects in the heart satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart. [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 define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as -structures on graded manifolds. The paper classifies -manifolds of maximal degree 3. [Copyright &y& Elsevier]

Abstract: In this paper, we apply a modification theorem for a compact homogeneous solvmanifold to compact complex homogeneous manifolds with pseudo-kählerian structures. We are able to classify these manifolds as certain products of projective rational homogeneous spaces, tori, simple and double reduced primitive pseudo-kähler spaces. [ABSTRACT FROM AUTHOR]

Abstract: Let S be a semigroup. In this paper we investigate the injectivity of as a Banach right module over . For weakly cancellative S this is the same as studying the flatness of the predual left module . For such semigroups S, we also investigate the projectivity of . We prove that for many semigroups S for which the Banach algebra is non-amenable, the -module is not injective. The main result about the projectivity of states that for a weakly cancellative inverse semigroup S, is projective if and only if S is finite. [Copyright &y& Elsevier]

Abstract: In this paper, we find the dimension and a method to obtain a basis of some infinitesimal deformations of the model Lie superalgebra . These deformations lie in being and the even and odd parts of the Lie superalgebra respectively. As corresponds to the model filiform Lie algebra, then these deformations can also be identified with the space of the infinitesimal deformations of the filiform -module . Combining with Bordemann et al. (2007) , Gómez et al. (2008)  and Khakimdjanov and Navarro  we therefore obtain a complete classification of all the infinitesimal deformations of the model Lie superalgebra . [Copyright &y& Elsevier]

Abstract: Consider the ring of Laurent polynomials in the variable τ. The Artin''s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ. In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We compute this cohomology for the cases , , , and with . [Copyright &y& Elsevier]

Abstract: In this paper, we describe the indices of the top and the least non-vanishing cohomologies of line bundles on Schubert varieties given by non-dominant weights in the Kac–Moody setting. We also prove some surjectivity results for maps between some cohomology modules. [Copyright &y& Elsevier]

Abstract: In this paper we present a cohomological description of the equivariant Brauer group relative to a Galois finite extension of fields endowed with the action of a group of operators. This description is a natural generalization of the classic Brauer–Hasse–Noether''s theorem, and it is established by means of a three-term exact sequence linking the relative equivariant Brauer group, the 2nd cohomology group of the semidirect product of the Galois group of the extension by the group of operators and the 2nd cohomology group of the group of operators. [Copyright &y& Elsevier]

We describe an algorithm for constructing a reasonably small CW-structure on the classifying space of a finite or automatic group G. The algorithm inputs a set of generators for G, and its output can be used to compute the integral cohomology of G. A prototype GAP implementation suggests that the algorithm is a practical method for studying the cohomology of finite groups in low dimensions. We also explain how the method can be used to compute the low-dimensional cohomology of finite crossed modules. The paper begins with a review of the notion of syzygy between defining relators for groups. This topological notion is then used in the design of the algorithm. [Copyright &y& Elsevier]

A classical theorem of Gromov states that the Betti numbers, i.e. the size of the free part of the homology groups, of negatively curved manifolds are bounded by the volume. We prove an analog of this theorem for the torsion part of the homology in all dimensions d ≠ 3. Thus the total homology is controlled by the volume. This applies in particular to the classical case of hyperbolic manifolds. In dimension 3 the size of torsion homology cannot be bounded in terms of the volume. As a byproduct, in dimension d ≥ 4 we give a fairly precise estimate for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension d ≥ 5 up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest. [ABSTRACT FROM AUTHOR]

We consider the -dimensional real superspace endowed with its standard contact structure defined by the 1-form . The conformal Lie superalgebra acts on as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra . We classify -invariant linear differential operators from to vanishing on , where is the superspace of bilinear differential operators between the superspaces of weighted densities. This result allows us to compute the first differential -relative cohomology of with coefficients in . This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on acting on bilinear differential operators, Int. J. Geom. Methods Mod. Phys. 2(1) (2005) 23-40]. [ABSTRACT FROM AUTHOR]

Leech's (co)homology groups of finite cyclic monoids are computed. [ABSTRACT FROM AUTHOR]

We construct linear maps from the spaces of quasimodular forms for a discrete subgroup Γ of SL(2,ℝ) to some cohomology spaces of the group Γ and prove that these maps are equivariant with respect to appropriate Hecke operator actions. The results are obtained by using the fact that there is a correspondence between quasimodular forms and certain finite sequences of modular forms. [ABSTRACT FROM AUTHOR]

We compute the first cohomology of the ortosymplectic Lie superalgebra 픬픰픭(1|2) on the (1, 1)-dimensional real superspace with coefficients in the superspace 픇λ, μ of bilinear differential operators acting on weighted densities (λ ∈ ℝ2 and μ ∈ ℝ). This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on ℝℙ1 acting on bilinear differential operators, Int. J. Geom. Meth. Mod. Phys.2(1) (2005) 23-40]. [ABSTRACT FROM AUTHOR]

We calculate cohomologies of the Schwartz algebras D and S, which are basic spaces (test-function spaces) of the distribution theory. In the process, we find special cohomologies of the quotient algebras E/D and M/S (E andMare also test-function spaces), which are quite unusual from the standpoint of the standard functional analysis and are interesting for theoretical and mathematical physics. [ABSTRACT FROM AUTHOR]

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651-683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651-683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225-255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold. [ABSTRACT FROM AUTHOR]

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge-de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian. [ABSTRACT FROM AUTHOR]

For a knot K with a framing F, we show that the knot rack R(K, F) of (K, F) has the structure of a rack extension of the knot quandle Q(K), via the projection canonically obtained by disregarding its framing. From this fact, we can construct a cohomology class [ϕF] of Q(K) corresponding to the framing: a cohomology class in $(H_\mathrm{R}^2(Q(K); \mathbb{Z})$ for K non-trivial, or one in $H_\mathrm{R}^2(Q(K); \mathbb{Z}/w_F\mathbb{Z})$ for K trivial, where wF denotes the writhe of (K, F). Moreover, when K is non-trivial, [ϕF] has a natural decomposition to the sum of $[\hat{\phi}_F] \in H_\mathrm{Q}^2(Q(K); \mathbb{Z})$ and $[\psi] \in H_\mathrm{D}^2(Q(K); \mathbb{Z})$, where [ψ] is independent to F. This correspondence $F \mapsto [\hat{\phi}_F]$ gives a bijection between the set of framings of K and $H_\mathrm{Q}^2(Q(K); \mathbb{Z})$. [ABSTRACT FROM AUTHOR]

Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q-pseudoconvex, a strongly q-pseudoconcave and two flat (i.e. locally zero sets of pluriharmonic functions) hypersurfaces. Finiteness and vanishing cohomology theorems obtained in [Saracco and Tomassini, Math. Z. 256: 737-748, 2007, Bull. Sci. Math. 132: 232-245, 2008] for semi q-coronae are generalized in this context and lead to results on extension problems and removable sets for sections of coherent sheaves and analytic subsets. [ABSTRACT FROM AUTHOR]