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R. Sazeedeh showed that top local cohomology modules are Gorenstein injective in a Gorenstein local ring with at most two dimensions. In this paper, it is proved that the condition of dimension in his result cannot be relaxed and the conclusion in his result holds for complete local hypersurface rings with any dimension. [ABSTRACT FROM AUTHOR]

In this paper, we show that rational homology 3-spheres are ubiquitous from the viewpoint of Heegaard splitting. Let M = H+ UF H_ be a genus g Heegaa;rd splitting of a closed 3-manifold and c be a simple closed curve in F. Then there is a 3-manifold Mc which is obtained from M by horizontal Dehn surgery along c. We show that for c such that the homology class [c] is generic in the set of curve-represented homology classes Hs ⊂ H1(F), rank(H1(Mc,Q)) < max{ 1, rank(H1(M,Q)}. As a corollary, for a set of cusub>, c2, hellip;, cm}, m ≥ g, such that each [ci] is generic in Hs ⊂ H1(F), M(c1,c2,…,cm) is a rational homology 3-sphere. [ABSTRACT FROM AUTHOR]

For a given endomorphism on a unitary k-algebra, A, with k in the center of A, there are definitions of twisted cyclic and Hochschild homology. This paper will show that the method used to define them can be used to define twisted de Rham homology. The main result is that twisted de Rham homology can be thought of as the kernel of the Connes map from twisted cyclic homology to twisted Hochschild homology. [ABSTRACT FROM AUTHOR]

A degeneration of compact Kähler manifolds gives rise to a monodromy action on the Betti moduli space H¹(X, G) = Hom(π1(x), G)/G over smooth fibres with a complex algebraic structure group G that is either abelian or reductive. Assume that the singularities of the central fibre are 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. [ABSTRACT FROM AUTHOR]

This paper records two results which were inexplicably omitted from the paper on Pin structures on low dimensional manifolds in the LMS Lecture Note Series, volume 151, by Kirby and this author. Kirby declined to be listed as a coauthor of this paper. par A $Pin^{-}$-structure on a surface $X$ induces a quadratic enhancement of the mod $2$ intersection form, $qcolon H_1(X;mathbb {Z}/2mathbb {Z})to mathbb {Z}/4mathbb {Z}$. par Theorem 1.1 says that $q$ vanishes on the kernel of the map in homology to a bounding $3$-manifold. This is used by Kreck and Puppe in their paper in Homology, Homotopy and Applications, volume 10. The arXiv version, arXiv:0707.1599 [math.AT], referred to an email from the author to Kreck for the proof. A more polished and public proof seems desirable. par In Section 6 of the paper with Kirby, a $Pin^{-}$-structure is constructed on a surface $X$ dual to $w_2$ in an oriented 4-manifold, $M^4$. Theorem 2.1 says that $q$ vanishes on the Poincaré dual to the image of $H^1(M;mathbb {Z}/2mathbb {Z})$ in $H^1(X;mathbb {Z}/2mathbb {Z})$. [ABSTRACT FROM AUTHOR]

We study twisted cohomology and homology groups on a onedimensional complex torus minus n distinct points with coefficients in a certain local system of rank one. This local system comes from the integrand of the Riemann-Wirtinger integral introduced by Mano. We construct bases of nonvanishing cohomology and homology groups, give an interpretation as a pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and finally describe briefly the Gauss-Manin connection on the cohomology groups. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold: first, we extend Saito's filtration on Chow groups, which is a candidate for the conjectural Bloch Beilinson filtration on the Chow groups of a smooth projective variety, from Chow groups to the bivariant Chow groups. In order to do this, we construct cycle class maps from the bivariant Chow groups to bivariant cohomology groups. Secondly, we use our methods to define a bivariant version of Bloch's higher Chow groups. [ABSTRACT FROM AUTHOR]

In this paper we define the 1,2-coloured HOMFLY-PT triply graded link homology and prove that it is a link invariant. We also conjecture on how to generalize our construction for arbitrary colours. [ABSTRACT FROM AUTHOR]

The Chern character maps are one of the most important working tools in mathematics. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this paper we eliminate this gap by characterizing the Chern character maps, from the Grothendieck group to the (negative) cyclic homology groups, in terms of simple universal properties. [ABSTRACT FROM AUTHOR]

In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of $C^infty (M)rtimes G$ for a finite group $G$ acting on a compact manifold $M$. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on $C^infty (M)rtimes G$ when $M$ is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures. [ABSTRACT FROM AUTHOR]

Let $G$ be the simple algebraic group $SL_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. In this paper, we compute the second cohomology of all irreducible representations of $G$. [ABSTRACT FROM AUTHOR]

Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle-dimensional primitive Hodge classes. In this paper, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle-dimensional cohomology classes, and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. [ABSTRACT FROM AUTHOR]

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite. [ABSTRACT FROM AUTHOR]

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and nullity. Although very elementary and easy to compute, this bound turns out to be suprisingly efficient. In particular, it makes it a routine check to recover the splitting number of 129 out of the 130 prime links with at most 9 crossings. Also, we easily determine 16 of the 17 splitting numbers that were studied by Batson and Seed using Khovanov homology, and later computed by Cha, Friedl and Powell using a variety of techniques. Finally, we determine the splitting number of a large class of 2-bridge links which includes examples recently computed by Borodzik and Gorsky using a Heegaard Floer theoretical criterion. [ABSTRACT FROM AUTHOR]

In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with Z[1/2]-coefficients of configuration spaces of odd-dimensional manifolds. [ABSTRACT FROM AUTHOR]

We compute the structure of the homology of an operad built from the spaces THd,n of configurations of points in Cd, modulo translation and homothety. We find that it is a mild generalization of Getzler's gravity operad, which occurs in dimension d = 1. [ABSTRACT FROM AUTHOR]

We consider the question: "If the zero-framed surgeries on two oriented knots in S³ are Z-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on K is Z-homology cobordant to the zero-framed surgery on many of its winding number one satellites P(K). Then we prove that in many cases the T and s-invariants of K and P(K) differ. Consequently neither T nor s is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the topological and smooth categories by proving similar results for K and its (p, 1)-cables. [ABSTRACT FROM AUTHOR]

We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill Zainoulline. The arguments used here are considerably simpler and allow us to remove all restrictions on the characteristic of the base field. [ABSTRACT FROM AUTHOR]

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincaré dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-selfdual SO(3) connections on a closed four-manifold. This should be a useful tool in comparing gauge theoretic invariants of smooth four-manifolds. [ABSTRACT FROM AUTHOR]

We prove a generalization of the theorem which has been proved by Fernandez, Ibanez, and de Leon. By this result, we give examples of non-Kähler manifolds which satisfy the property of compact Kähler manifolds concerning the coeffective cohomology. [ABSTRACT FROM AUTHOR]

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology. [ABSTRACT FROM AUTHOR]

We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the Pontryagin product on the equivariant homology of the based loop group ΩK. [ABSTRACT FROM AUTHOR]

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the non-simply connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology. [ABSTRACT FROM AUTHOR]

Let k be a perfect field of virtual cohomological dimension ≤ 2. Let G be a connected linear algebraic group over k such that Gsc satisfies a Hasse principle over k. Let X be a principal homogeneous space under G over k. We show that if X admits a zero cycle of degree one, then X has a k-rational point. [ABSTRACT FROM AUTHOR]

The article presents a mathematical study on the finite generation of Tate cohomology (TC) of a finite group with coefficients in a indecomposable nonprojective module. It is proved that if the finite group has periodic cohomology then all finitely generated modules over the finite group finitely generate TC. Also mentioned is a conjecture on the equivalence of support variety of the module and the maximal ideals. Several theorems and their proofs related to the study are also presented.

The article discusses Heegaard Floer homology which is an oriented three-manifolds that use Heegaard diagrams of the three-manifold. It states that the construction of a suitable variant of Lagrangian Floer homology is in symmetry of a Heegaard surface. It notes that the null-homologous knot Floer homology detects the genus of the knot and are generalized to the case of links which give a multi-graded Abelian group.

The article presents the mathematic proof of Thurston norm of symplectic S1 multiplied by N1 3 and subgroup separability. It shows the mathematical formula to show symplectic 4-manifolds and a class of 3-manifolds whose fundamental group satisfies appropriate subgroup separability conditions. It provides the Seiberg-Witten proof of Thurston theory wherein twisted homology and Alexander modules are defined through the Mayer-Vietoris type exact sequence.

The article examines the classical relation between the invariant theory of the special linear group (SL(r)) and cohomology of Grassmannians (Gr(r,n)). It highlights how triple intersections of Schubert varieties produce geometrically a basis of the invariants space for the related SL(r) tensor product multiciplicity problem. It discusses the Witten's relation between structure coefficients and the quantum cohomology of Grassmannians in the Verlinde algebra.