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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]

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]

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]

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]

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]