Pro-p completions of Poincaré duality groups

We consider some sufficient conditions for the pro-p completion of an orientable Poincaré duality group of dimension n ≥ 3 to be a virtually pro-p Poincaré duality group of dimension at most n ? 2.     

Finite automorphisms of negatively curved Poincaré duality groups

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a -hyperbolic Poincaré Duality group over , then the fixed subgroup is a Poincaré Duality group over . We also provide a family of examples to show that the fixed subgroup might not be a Poincaré Duality group over . In fact, the fixed subgroups in our examples even fail to be duality groups over .     

Poincaré duality for algebraic de rham cohomology

We discuss in some detail the algebraic notion of De Rham cohomology with compact supports for singular schemes over a field of characteristic zero. We prove Poincaré duality with respect to De Rham homology as defined by Hartshorne [H.75], so providing a generalization of some results of that paper to the non proper case. In order to do this, we work in the setting of the categories introduced by Herrera and Lieberman [HL], and we interpret our cohomology groups as hyperext groups. We exhibit canonical morphisms of cospecialization from complex-analytic De Rham (resp. rigid) cohomology groups with compact supports to the algebraic ones. These morphisms, together with the specialization morphisms [H.75, IV.1.2] (resp. [BB, 1]) going in the opposite direction, are shown to be compatible with our algebraic Poincaré pairing and the analogous complex-analytic (resp. rigid) one (resp. [B.97, 3.2]).Mathematics Subject Classification (2000):Primary 14FXX     

Combinatorial cell complexes and Poincaré duality

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c’s and develop enough algebraic topology in this setting to prove the Poincaré duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.     

Poincaré duality isomorphisms in tensor categories

If for a vector space V of dimension g over a characteristic zero field we denote by ${\wedge }^{i}V$ its alternating powers, and by $V^{\vee }$ its linear dual, then there are natural Poincaré isomorphisms:

${\wedge }^i{V}^{\vee }?{\wedge }^{g?i}V.$

We describe an analogous result for objects in rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories.     

Poincaré duality for transitive unimodular invariantly oriented Lie algebroids

Fubini's formula for an oriented bundle suggests a definition of the integration of forms of maximal degree in a transitive Lie algebroid A (using the fibre integral ∫_A in A, defined and investigated by the author in his previous paper). In the case of a unimodular and invariantly oriented transitive Lie algebroid, this integral enables us to define the Poincaré scalar product. The purpose of this paper is to investigate the fundamental properties of this product.     

Mackey Theory for p-adic Lie Groups

This paper gives a p-adic analogue of the Mackey theory, which relates representations of a group of type G - H × t A to systems of imprimitivity.     

Classification of Poincaré duality algebras over the rationals

The Poincaré duality algebras over Q play a key role in the rational homotopy classification of closed manifolds [3]. In this paper we give a way of classifying general Poincaré duality algebras and then specialize to the case of algebras which are generated by some homogeneous component and show how the classification reduces to the linear classification of certain homogeneous polynomials and exterior forms.     

Uniform K-theory,and Poincaré duality for uniform K-homology

We revisit ?pakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology.We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with uniform K-homology and prove Poincaré duality between uniform K-theory and uniform K-homology on spin^c manifolds of bounded geometry.     

Poincaré duality and periodicity,II. James periodicity

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension Σ ^j K so that the resulting space satisfies Poincaré duality. When this is possible, we say that Σ ^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then Σ ^j K is a spine whenever j is a suitable power of two.     

Moritaʼs duality for split reductive groups

In this paper, we extend the work in [Z. Qi, C. Yang, Morita?s theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 [7]] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations.     

Every finite complex is the classifying space for proper bundles of a virtual Poincaré duality group

We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.     

Poincaré Duality for Logarithmic Crystalline Cohomology

We prove Poincaré duality for logarithmic crystalline cohomology of log smooth schemes whose underlying schemes are reduced. This is a generalization of the result of P. Berthelot for usual smooth schemes and that of O. Hyodo for the special fibers of semi-stable families and trivial coefficients.     

Fractional Poincaré inequalities for general measures

We prove a fractional version of Poincaré inequalities in the context of ${\mathbb{R}}^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.