Related papers: Superforms, Tropical Cohomology, and Poincar\'e Du…

We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincar\'e duality and behaves analogously to the…

We prove a Poincar\'e duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincar\'e dual to each other. This allows us to define…

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 tropical…

We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincar\'e duality. The condition is always satisfied when the…

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a…

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincar\'e…

We construct an explicit de Rham isomorphism relating the cohomology rings of Banagl's de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities. Intersection space…

Intersection homology with coefficients in a field restores Poincar\'e duality for some spaces with singularities, as pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and pseudomanifolds are different. This…

We investigate the problem of Poincar\'e duality for $L^p$ differential forms on bounded subanalytic submanifolds of $\mathbb{R}^n$ (not necessarily compact). We show that, when $p$ is sufficiently close to $1$ then the $L^p$ cohomology of…

Using a cap product, we construct an explicit Poincar\'e duality isomorphism between the blown-up intersection cohomology and the Borel-Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented…

We define monodromy maps for tropical Dolbeault cohomology of algebraic varieties over non-Archimedean fields. We propose a conjecture of Hodge isomorphisms via monodromy maps, and provide some evidence.

We extend Poincar\'e duality in \'etale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.

We use piecewise polynomials to define tropical cocycles generalising the well-known notion of tropical Cartier divisors to higher codimensions. Groups of cocycles are tropical analogues of Chow cohomology groups. We also introduce an…

We prove that $p$-adic geometric pro-\'etale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincar\'e duality as we have conjectured. This…

For having a Poincar\'e 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…

Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality…

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies…

We continue our study of the variation of parabolic cohomology (math.AG/0310139) and derive an exact formula for the underlying Poincare duality. As an illustration of our methods, we compute the monodromy of the Picard-Euler system and its…

We introduce real-valued $(p,q)$-forms on weighted metric graphs with boundary similar to Lagerberg forms on polyhedral spaces. We compute the Dolbeault cohomology and prove Poincar\'e duality. Using Thuillier's thesis, the skeleton of a…

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…