The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, ...the fan is said to be a
. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a
In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.
We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincaré duality ...for cohomology of tropical manifolds, which are polyhedral spaces locally given by Bergman fans of matroids.
We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum after looping. This takes ...inspiration from a recent work of Jeffrey and Selick, in which they study pullback fibrations of this type but under stronger hypotheses compared to our result.
We study the “twisted” Poincaré duality of smooth Poisson manifolds, and show that, if the modular vector field is diagonalizable, then there is a mixed complex associated to the Poisson complex, ...which, combining with the twisted Poincaré duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology. This generalizes the previous results obtained by Xu for unimodular Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure is preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras it is also preserved by Koszul duality.
A version of the twisted Poincaré duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by ...twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincaré duality reduces to the Poincaré duality in usual sense. The main result generalizes the work of Launois and Richard 8 for the quadratic Poisson structures and Zhu 25 for the linear Poisson structures.
A note on Gorenstein spaces Félix, Yves; Halperin, Steve
Journal of pure and applied algebra,
November 2019, 2019-11-00, Letnik:
223, Številka:
11
Journal Article
Recenzirano
Associated with an augmented differential graded algebra R=R≥0 is a homotopy invariant T(R). This is a graded vector space, and if H0(R) is the ground field and H>N(R)=0 then dimT(R)=1 if and only if ...H(R) is a Poincaré duality algebra. In the case of Sullivan extensions ∧W→∧W⊗∧Z→∧Z in which dimH(∧Z)<∞ we show thatT(∧W⊗∧Z)=T(∧W)⊗T(∧Z). This is applied to finite dimensional CW complexes X where the fundamental group G acts nilpotently in the cohomology H(X˜;Q) of the universal covering space. If H(X;Q) is a Poincaré duality algebra and H(X˜;Q) and H(BG;Q) are finite dimensional then they are also Poincaré duality algebras.
Poincaré duality and resonance varieties Suciu, Alexander I.
Proceedings of the Royal Society of Edinburgh. Section A. Mathematics,
12/2020, Letnik:
150, Številka:
6
Journal Article
Recenzirano
Odprti dostop
We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric ...description of the resonance varieties ${\cal R}^i_k(A)$.
Singular decompositions of a cap product Chataur, David; Saralegi-Aranguren, Martintxo; Tanré, Daniel
Proceedings of the American Mathematical Society,
08/2017, Letnik:
145, Številka:
8
Journal Article
Recenzirano
Odprti dostop
In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection ...homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)homology that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincaré duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities.
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