No description available.

No description available.

No description available.

No description available.

It is known that some discrete semigroup algebras have trivial continuous simplicial cohomology, at least in high dimensions. The aim of this work is to investigate the situation for the locally compact case, which even for the important example of the positive real numbers is not clear.

In this thesis we study the second Hochschild cohomology group HH 2(Lambda) of a finite dimensional algebra Lambda. In particular, we determine HH2(Lambda) where Lambda is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field K and show that this group is zero for most such Lambda; we give a basis for HH2(Lambda) in the few cases where it is not zero.;Then we consider algebras of tame representation type; more specifically, we study finite dimensional self-injective one parametric tame algebras which are not weakly symmetric. Here we show that HH2(Lambda) is non-zero and find a non-zero element eta in HH2(Lambda) and an associative deformation Lambdaeta of Lambda.

In this thesis we investigate groups whose nth cohomology functors commute with filtered colimits for all sufficiently large n. In Chapter 1 we introduce some basic definitions and important background material. We make the definition that a group G has cohomology almost everywhere finitary if and only if the set F(G) of natural numbers n for which the nth cohomology of G commutes with filtered colimits is cofinite. We also introduce Kropholler's class LHF of locally hierarchically decomposable groups. We then state a key result of Kropholler, which establishes a dichotomy for this class: If G is an LHF-group, then the set F(G) is either finite or cofinite. Kropholler's theorem does not, however, give a characterisation of the LHF-groups with cohomology almost everywhere finitary, and this is precisely the problem that we are interested in. In Chapter 2 we investigate algebraic characterisations of certain classes of LHF-groups with cohomology almost everywhere finitary. In particular, we establish sufficient conditions for a group in the class H1F to have cohomology almost everywhere finitary. We prove a stronger result for the class of groups of finite virtual cohomological dimension over a ring R of prime characteristic p, and use this result to answer an open question of Leary and Nucinkis. We also consider the class of locally (polycyclic-by-finite) groups, and show that such a group G has cohomology almost everywhere finitary if and only if G has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of G is finitely generated. We then change direction in Chapter 3, and show an interesting connection between this problem and the problem of group actions on spheres. In particular, we show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere. Finally, in Chapter 4 we provide a topological characterisation of the LHF-groups with cohomology almost everywhere finitary. In particular, we show that if G is an LHF-group with cohomology almost everywhere finitary, then GxZ has an Eilenberg-Mac Lane space K(GxZ,1) with finitely many n-cells for all sufficiently large n. It is an open question as to whether the LHF restriction can be dropped here. We also show that the converse statement holds for arbitrary G.

Cette thèse se concentre sur l'homotopie rationnelle des espaces d'intersection, espaces définis et développés par M. Banagl, et se décompose en trois parties :La première partie traite de la dualité de Poincaré associée aux espaces d'intersection. Étant donnée X une pseudovariété compacte, connexe et orientée de dimension n=4s à singularités isolées, les espaces d'intersections associés aux deux perversités milieux coïncident ImX ≈ InX. Cela nous permet de définir une forme d'intersection bilinéaire symétrique non dégénérée bHI : H2s(ImX) x H2s(ImX) ---> Q provenant d'une dualité de Poincaré généralisée définie sur l'homologie rationnelle des espaces d'intersection. Cette dualité ne provient pas de l'évaluation d'un cup produit contre une classe fondamentale. En utilisant le formalisme des espaces d'intersection nous montrons, dans le cas de la dimension paire, qu'il est possible de construire un espace à dualité de Poincaré rationnelle DP(X). Lorsque dim X = 4s la classe de Witt associée à la forme d'intersection bDP(X), définie via dualité de Poincaré, est la même que la classe Witt de bHI dans le groupe W(Q). Nous montrons aussi comment construire de tels espaces DP(X) de le cas d'une dimension impaire.La second partie développe la notion d'espace d'intersection lagrangien, notion introduite dans le premier chapitre pour construire DP(X) lorsque dim X = 2s+1. Nous montrons que l'homologie rationnelle de ces espaces interagit avec les homologies d'intersection milieu IHm(X) et IHn(X) au travers d'un diagramme commutatif que nous appelons un diagramme (s+1,s)-biréflexif. Dans une seconde partie, nous montrons que la notion de troncation homologique utilisée pour définir les espaces d'intersection peut être rendue fonctorielle lorsque l'on se concentre sur les espaces rationnels nilpotents de type fini.Pour finir, la troisième partie étudie l'interaction entre la théorie de Hodge mixte et la cohomologie rationnelle des espaces d'intersection pour X une variété algébrique projective complexe à singularités isolées. Nous montrons que la cohomologie de ces espaces d'intersection possède de façon naturelle une structure de Hodge mixte définie au niveau des modèles rationnels. Ces structures de Hodge mixte nous permettent alors de déduire des résultats sur la formalité des espaces d'intersection. / This thesis is concerned with the rational homotopy theory of intersection spaces. It is composed of three parts, each of them being more or less independent. The first part concerns the notion of Poincaré duality associated to the intersection spaces. When X is a compact connected oriented pseudomanifold of dimension $n=4s$ with only isolated singularities, we then have a well defined middle perversities intersection spaces ImX ≈ InX. with a non degenerate symmetric intersection form bHI : H2s(ImX) x H2s(ImX) ---> QThis intersection form comes from a generalized Poincaré duality defined on intersection spaces, but is not defined as the evaluation of a cup product against a fundamental class. We construct rational Poincaré duality spaces DP(X) such that when dim X =4s the Witt class of the intersection form bDP(X), associated to DP(X) is the same that bHI in the Witt group W(Q). We also show how to construct Poincaré duality spaces DP(X) when n =2s+1.The second part develop the notion of Lagrangian intersection spaces introduced in the first part to construct DP(X) when dim X =2s+1. We show that the rational homology of these spaces lies between the two middle perversities intersection homology IHm(X) and IHn(X) in a sense that we call a (s+1,s)-bireflective diagram. In a second section we show that the notion of homology truncation can be made functorial when we deal with nilpotent rational spaces of finite type.The last part is devoted to the interaction between Hodge theory and the rational cohomology of intersection spaces when X is a complex projective algebraic varieties with only isolated singularities. We show that theses spaces carry a natural mixed Hodge structure at the algebraic models level. We then use these mixed Hodge structures to derive results about the formality of intersection spaces.

Une difféologie sur un ensemble arbitraire X, déclare, pour tout entier n,quelles applications de R[exposant n] vers X sont lisses. Cette idée est structurée par trois axiomes naturels : recouvrements, localité et compatibilité lisse. L’un des objectifs de cette thèse est de développer et d’étudier des outils classiques de la topologie algébrique dans le cadre difféologique. Parmi ces outils on se penche particulièrement sur les théories homologiques et cohomologiques généralisées. Un autre objectif est de montrer que les espaces difféologiques offrent un cadre assez naturel afin d’étudier les espaces singuliers : pseudo-variétés contrôlées à la Thom-Mather. On met en place les définitions de théories (co)homologiques généralisées dans la catégorie Diff . On définit une nouvelle notion "CW-difféologie" liée à la notion de CW-complexes. P. Iglesias Zemmour a introduit l’homologie cubique et cohomologie de De Rham dans la cadre difféologique. On développe en outre l’homologie singulière, l’homologie cellulaire et la cohomologie de Rham difféologique. On étudie les pseudo-variétés contrôlées qui sont des espaces singuliers en difféologie. / A diffeology on an arbitrary set X declares, for any integer n, which applications in R[exponent n] to X are smooth. This idea is structured by three natural axioms covering, locality and smooth compatibility. One objective of this thesis is to develop and study classical tools of algebraic topology in the diffeological framework. These tools are particularly looking at the generalized homology and cohomology theories. Another objective is to show that diffeological spaces offer a fairly natural frame to study the singular spaces : Thom-Mather stratified space. We set up the definitions of generalized (co)homology theories in the category Diff. We define a new notion of " CW- diffeology " linked to the notion of CW- complexes. P.Iglesias Zemmour introduced cubic homology and De Rham cohomology in the diffeological framework. We develop in addition the singular homology, cellular homology and diffeological de Rham cohomology. We study Thom-Mather stratified spaces which are singular spaces, with diffeology.

Dans son article de 1966 intitulé "Ktheory and Reality", Atiyah introduit une variante de la Kthéorie des fibres vectoriels complexes, notée KR, qui, d'une certaine manière, englobe à la fois la Ktheory complexe KU, la Ktheory réelle KO (dite aussi orthogonale), et la Kthéorie autoconjuguée KSc d'Anderson. Dans cette thèse, nous généralisons cette théorie au cadre noncommutatif de la Kthéorie tordue des groupoïdes topologiques. Nous développons ainsi la KRthéorie tordue des groupoïdes en nous servant principalement des outils de la KKthéorie "réelle" de Kasparov. Il s'agit notamment de l'étude de la Kthéorie des C*algèbres graduées associées à des systèmes dynamiques de groupoides munis de certaines involutions. Les classes d'équivalence de tels systèmes composent le groupe de Brauer Réel gradué que nous définissons et calculons en termes de classes de cohomologie de Cech. Nous donnons dans cette nouvelle théorie les analogues des résultats classiques en Kthéorie tels que les suites exactes de MayerVietoris, la périodicité de Bott et le théorème d'isomorphisme de Thom / In his 1966's paper "Ktheory and Reality", Atiyah introduced a variant of Ktheory of complex vector bundles called KRtheory, which, in some sense, is a mixture of complex Ktheory KU, real Ktheory (also called orthogonal Ktheory) KO, and Anderson's selfconjugate Ktheory KSc. The main purpose of this thesis is to generalize that theory to the noncommutative framework of twisted groupoid Ktheory. We then introduce twisted groupoid KRtheory by using the powerful machineries of Kasparov's "real" KKtheory. Specifically, we deal with the Ktheory of graded C*algebras associated with groupoid dynamical systems endowed with involutions. Such dynamical systems are classified by the Real graded Brauer group to be defined and computed in terms of Cech cohomology classes. In this new Ktheory, we give the analogues of the fundamental results in Ktheory such as the MayerVietoris exact sequences, the Bott periodicity and the Thom isomorphism theorem