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 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 Refined intersection homology on non-Witt spaces

2014 ◽  
Vol 07 (01) ◽  
pp. 105-133 ◽  
Author(s):  
Pierre Albin ◽  
Markus Banagl ◽  
Eric Leichtnam ◽  
Rafe Mazzeo ◽  
Paolo Piazza
Keyword(s):  
Homology Theory ◽  
Cohomology Theory ◽  
The Self ◽  
Intersection Cohomology ◽  
Intersection Homology ◽  
Stratified Spaces ◽  
Constructible Sheaves ◽  
De Rham Theory ◽  
De Rham ◽  
Definition Of

We investigate a generalization to non-Witt stratified spaces of the intersection homology theory of Goresky–MacPherson. The second-named author has described the self-dual sheaves compatible with intersection homology, and the other authors have described a generalization of Cheeger's L2 de Rham cohomology. In this paper we first extend both of these cohomology theories by describing all sheaf complexes in the derived category of constructible sheaves that are compatible with middle perversity intersection cohomology, though not necessarily self-dual. Our main result is that this refined intersection cohomology theory coincides with the analytic de Rham theory on Thom–Mather stratified spaces. The word "refined" is motivated by the fact that the definition of this cohomology theory depends on the choice of an additional structure (mezzo-perversity) which is automatically zero in the case of a Witt space.

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 An approach to intersection theory on singular varieties using motivic complexes

2016 ◽  
Vol 152 (11) ◽  
pp. 2371-2404 ◽  
Author(s):  
Eric M. Friedlander ◽  
J. Ross
Keyword(s):  
Equivalence Relation ◽  
Homology Theory ◽  
Intersection Theory ◽  
Equivalence Classes ◽  
Cohomology Theory ◽  
Intersection Homology ◽  
Localization Property ◽  
Singular Varieties

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

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.

Sign in / Sign up

Export Citation Format

Share Document