defined by the cap product with the fundamental class [M ] of M is an isomorphism of finitely generated abelian groups. This duality is manifest in many ways in geometry. Locally it corresponds to the duality of forms and currents in the theory of de Rham. It appears in Hodge theory via the *operator on harmonic forms. In Poincaré’s original work it can be described via barycentric subdivision ...

We first present a short review of general supersymmetric compactifications in string and M-theory using the language of G-structures and intrinsic torsion. We then summarize recent work on the generic conditions for supersymmetric AdS5 backgrounds in M-theory and the construction of classes of new solutions. Turning to AdS5 compactifications in type IIB, we summarize the construction of an inf...

In this article we investigate the relations between Gorenstein projective dimensions of [Formula: see text]-modules and their socles for text]-minimal Auslander–Gorenstein algebras text]. First give a description projective-injective in terms socles. Then prove that text]-module text] has dimension at most if only its socle is cogenerated by text]-module. Furthermore, show can be characterised...

It is well known that the controllability of a linear multidimensional control system depends on an algebraic property (namely, the torsion-freeness) of a certain module M associated with the system (Oberst, 1990; Pillai and Shankar, 1999; Pommaret and Quadrat, 1999a; 1999b). The recent survey (Wood, 2000) gives different equivalent formulations of controllability and, in particular, the equiva...

The moduli space of complex structures on a compact Riemann surface of genus 1 or ≥ 2 can be identified with the deformation space of Riemannian metrics of constant curvature 0 or −1 respectively, while the latter definition naturally gives rise to the Weil-Peterson metric. Let M be a compact, oriented, and spin manifold of dimension 7. Then M admits a differential 3-form φ of generic type call...

We present a gauge theory of the conformal group in four spacetime dimensions with non-vanishing torsion. In particular, we allow for completely antisymmetric torsion, equivalent by Hodge duality to an axial vector whose presence does not spoil invariance theory, contrast claims antecedent literature. The requirement implies differential condition (in Killing equation) on aforementioned which l...

We prove that the Ozsváth-Szabó contact invariant of a closed contact 3manifold with positive 2π–torsion vanishes. In 2002, Ozsváth and Szabó [OSz1] defined an invariant of a closed contact 3-manifold (M, ξ) as an element of the Heegaard Floer homology group ĤF (−M). The definition of the contact invariant was made possible by the work of Giroux [Gi3], which related contact structures and open ...

Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M , under certain technical conditions on μ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geomet...

We use the one-parameter fixed point theory of Geoghegan and Nicas to get information about the closed orbit structure of transverse gradient flows of closed 1-forms on a closed manifold M . We define a noncommutative zeta function in an object related to the first Hochschild homology group of the Novikov ring associated to the 1-form and relate it to the torsion of a natural chain homotopy equ...