Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular ...derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a Gerstenhaber algebra admits a Batalin-Vilkovisky operator inherited from some one of its Poisson cochain complex if and only if the Poisson structure is pseudo-unimodular. This generalizes the geometric version due to P. Xu. The modular derivation and Batalin-Vilkovisky operator are also described by using the dual basis of the Kähler differential module.
Poincaré duality for posets Macías-Virgós, E.; Mosquera-Lois, D.; Vilches, J.A.
Topology and its applications,
11/2023, Letnik:
339
Journal Article
Recenzirano
Odprti dostop
The main goal of this paper is to prove a Poincaré duality theorem in the context of finite posets. This result will be established for the class of homologically bi-admissible finite posets, which ...includes the well-known subclass of finite closed homology manifolds.
We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of ...tropical homology, and we show that it behaves analogously to classical Borel-Moore homology in the sense that there are proper push-forwards, cross products, and cup products with tropical cohomology classes, and that it satisfies identities like the projection formula and the Künneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we characterize the rational polyhedral spaces that satisfy Poincaré-Verdier duality as those that are smooth.
A short note on simplicial stratifications Wrazidlo, Dominik
Proceedings of the American Mathematical Society. Series B,
5/2023, Letnik:
10, Številka:
17
Journal Article
Recenzirano
We show that the simplicial stratification associated to a triangulation of a PL pseudomanifold possesses a canonical system of trivializations of link bundles that satisfies a natural compatibility ...condition over nested singular strata. Consequently, Agustín Vicente and Fernández de Bobadilla’s generalization of Banagl’s intersection space construction is applicable to all PL pseudomanifolds (and in particular, to all complex algebraic varieties).
For having a Poincaré duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure ...need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex which gives a Poincaré duality via a cap product with intersection homology, for any commutative ring of coefficients. We prove also the topological invariance of the blown-up intersection cohomology with compact supports in the case of a paracompact pseudomanifold with no codimension one strata.
This work is written with general perversities, defined on each stratum and not only in function of the codimension of strata. It contains also a tame intersection homology, suitable for large perversities.
We prove the Poincaré duality theorem for bi-cellular posets X, that is, both X and Xop are cellular, in terms of cap product for finite posets which will be introduced. Moreover, we show that our ...results include face posets of h-regular homology manifolds.
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