scholarly journals Poincaré duality for $$L^p$$ cohomology on subanalytic singular spaces

2021 ◽  
Author(s):  
Guillaume Valette
Keyword(s):  
Differential Forms ◽  
Intersection Homology ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Singular Spaces ◽  
Singular Homology

AbstractWe investigate the problem of Poincaré duality for $$L^p$$ L p differential forms on bounded subanalytic submanifolds of $$\mathbb {R}^n$$ R n (not necessarily compact). We show that, when p is sufficiently close to 1 then the $$L^p$$ L p cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that $$L^p$$ L p cohomology is dual to intersection homology. As a consequence, we can deduce that the $$L^p$$ L p cohomology is Poincaré dual to $$L^q$$ L q cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.

scholarly journals Intersection space cohomology of three-strata pseudomanifolds

2019 ◽  
pp. 1-50
Author(s):  
J. Timo Essig
Keyword(s):  
Mirror Symmetry ◽  
Differential Forms ◽  
Main Property ◽  
Homology Theory ◽  
Cohomology Theory ◽  
Intersection Homology ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Cell Complexes ◽  
Space Cohomology

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 Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.

scholarly journals Variations on Poincaré duality for intersection homology

2020 ◽  
Vol 65 (1) ◽  
pp. 117-154
Author(s):  
Martintxo Saralegi-Aranguren ◽  
Daniel Tanré
Keyword(s):  
Intersection Homology ◽  
Poincaré Duality ◽  
Poincare Duality

scholarly journals POINCARÉ DUALITY, CAP PRODUCT AND BOREL–MOORE INTERSECTION HOMOLOGY

2020 ◽  
Vol 71 (3) ◽  
pp. 943-958
Author(s):  
Martintxo Saralegi-Aranguren ◽  
Daniel Tanré
Keyword(s):  
Commutative Ring ◽  
Intersection Cohomology ◽  
Intersection Homology ◽  
Poincaré Duality ◽  
Poincare Duality

Abstract Using a cap product, we construct an explicit Poincaré duality isomorphism between the blown-up intersection cohomology and the Borel–Moore intersection homology, for any commutative ring of coefficients and second-countable, oriented pseudomanifolds.

scholarly journals Poincaré duality with cap products in intersection homology

2018 ◽  
Vol 326 ◽  
pp. 314-351 ◽  
Author(s):  
David Chataur ◽  
Martintxo Saralegi-Aranguren ◽  
Daniel Tanré
Keyword(s):  
Intersection Homology ◽  
Poincaré Duality ◽  
Poincare Duality

scholarly journals TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

2021 ◽  
pp. 1-8
Author(s):  
DANIEL KASPROWSKI ◽  
MARKUS LAND
Keyword(s):  
Fundamental Group ◽  
Homotopy Equivalent ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Canonical Map ◽  
Duality Space ◽  
Simply Connected Manifolds ◽  
Simply Connected ◽  
Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

scholarly journals Tautological rings of spaces of pointed genus two curves of compact type

2016 ◽  
Vol 152 (7) ◽  
pp. 1398-1420 ◽  
Author(s):  
Dan Petersen
Keyword(s):  
Hodge Structure ◽  
Galois Representation ◽  
Mixed Hodge Structure ◽  
Poincaré Duality ◽  
General Study ◽  
Compact Type ◽  
Poincare Duality ◽  
Tautological Ring ◽  
Genus Two ◽  
Tautological Rings

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

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