scholarly journals On the L2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

2016 ◽  
Vol 08 (01) ◽  
pp. 151-186 ◽  
Author(s):  
Francesco Bei
Keyword(s):  
Riemannian Manifolds ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Finite Dimensional Vector Space ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Rham Complex ◽  
De Rham ◽  
Hilbert Complex ◽  
Incomplete Riemannian Manifolds

Let [Formula: see text] be an open, oriented and incomplete Riemannian manifold of dimension [Formula: see text]. Under some general conditions we show the existence of a Hilbert complex [Formula: see text] such that its cohomology groups, labeled with [Formula: see text], satisfy the following properties: [Formula: see text] [Formula: see text] (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: [Formula: see text] If [Formula: see text] is a Fredholm complex, then every closed extension of the de Rham complex [Formula: see text] is a Fredholm complex and, for each [Formula: see text], the quotient [Formula: see text] is a finite dimensional vector space.

scholarly journals Groups which satisfy a weak form of Poincaré duality

1991 ◽  
Vol 34 (3) ◽  
pp. 463-486
Author(s):  
J. E. Roberts
Keyword(s):  
Finite Rank ◽  
Weak Form ◽  
Homological Dimension ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Soluble Groups ◽  
Finite Dimensional ◽  
Duality Property ◽  
The Given ◽  
Torsion Free

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

scholarly journals Chiral de Rham Complex on Riemannian Manifolds and Special Holonomy

2013 ◽  
Vol 318 (3) ◽  
pp. 575-613 ◽  
Author(s):  
Joel Ekstrand ◽  
Reimundo Heluani ◽  
Johan Källén ◽  
Maxim Zabzine
Keyword(s):  
Riemannian Manifolds ◽  
De Rham Complex ◽  
Special Holonomy ◽  
Rham Complex ◽  
De Rham

scholarly journals MODULES FOR A SHEAF OF LIE ALGEBRAS ON LOOP MANIFOLDS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250079
Author(s):  
YULY BILLIG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Semidirect Product ◽  
Central Extension ◽  
Vector Fields ◽  
Vertex Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Rham Complex ◽  
De Rham

We consider a semidirect product of the sheaf of vector fields on a manifold ℂ* × X with a central extension of the sheaf of Lie algebras of maps from ℂ* × X into a finite-dimensional simple Lie algebra, viewed as sheaves on X. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov–Schechtman–Vaintrob on the chiral de Rham complex.

ABSTRACT AND CLASSICAL HODGE–DE RHAM THEORY

2012 ◽  
Vol 10 (01) ◽  
pp. 91-111 ◽  
Author(s):  
NAT SMALE ◽  
STEVE SMALE
Keyword(s):  
Riemannian Manifolds ◽  
Metric Spaces ◽  
Differential Forms ◽  
Small Scale ◽  
Smooth Functions ◽  
Compact Metric Spaces ◽  
Rham Complex ◽  
De Rham Theory ◽  
De Rham ◽  
Chain Maps

In previous work, with Bartholdi and Schick [1], the authors developed a Hodge–de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

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 Integral calculus on quantum exterior algebras

2014 ◽  
Vol 11 (04) ◽  
pp. 1450026 ◽  
Author(s):  
Serkan Karaçuha ◽  
Christian Lomp
Keyword(s):  
Differential Calculus ◽  
Polynomial Algebra ◽  
Exterior Algebra ◽  
Integral Calculus ◽  
De Rham Complex ◽  
Integral Forms ◽  
Rham Complex ◽  
De Rham ◽  
Quantum Polynomial

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

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