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Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $\Gamma$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with positive scalar curvature defined by Piazza and Schick corresponds to the $\varrho$-classes defined by the author of this paper.

An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0⟶Y⟶Z⟶X⟶0. When properly organized all these extensions constitute a linear space denoted by ExtB(X,Y), where B is the underlying (Banach) algebra. In this paper we "compute" the spaces of extensions for the Schatten classes when they are regarded in its natural (left) module structure over B=B(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows

Let p>3 be a prime number and K a finite extension of Q_p. We consider a proper and smooth surface XK over K, with a semistable model X over the ring of integers OK of K. In this thesis, we give a criterion for the good reduction of XK for the case of K3 surfaces, in terms of the monodromy operator in the second De Rham cohomology group. We don't use trascendental methods nor p-adic Hodge Theory as in other works concerning this problem. Instead, we first get a p-adic version of the Clemens-Schmid exact sequence and use it to study the degree of nilpotency of the monodromy operator N on the log-crystalline cohomology group of the special fiber Xs of the semistable model X. By the work of Nakkajima, we can assume that Xs is a combinatorial K3 surface. Then, we prove that Xs is of type I iff N=0; Xs if of type II iff N is not zero and N^2=0; Xs is of type III iff N^2 is not zero. In particular, this implies that XK has good reduction if and only if the monodromy operator on its second De Rham cohomology is zero. Finally, we also give some ideas on how to address the same problem for the case of Enriques surfaces. In particular, we prove that we are reduced to the case of K3 surfaces.

Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in KK-theory, we endow the K-groups of the arithmetic manifold $X/\Gamma$, of the reduced group C*-algebra of $\Gamma$ and of the boundary crossed product algebra associated to the action of $\Gamma$ on $\partial X$, with Hecke operators. The K-theory and K-homology groups of these C*-algebras are related by a Gysin six-term exact sequence. In the case when $\Gamma$ is a group of real hyperbolic isometries, we show that this Gysin sequence is Hecke equivariant. Finally, in the case when $\Gamma$ is a subgroup of a Bianchi group, we construct explicit Hecke-equivariant maps between the integral cohomology of $\Gamma$ and each of these K-groups. Our methods apply to torsion-free finite index subgroups of $PSL(2,\mathbb{Z})$ as well. These results are achieved in the context of unbounded Fredholm modules, shedding light on noncommutative geometric aspects of the purely infinite boundary crossed product algebra., Comment: 50 pages

The main result of this article is that any braided (resp. annular, planar) diagram group $D$ splits as a short exact sequence $1 \to R \to D \to S \to 1$ where $R$ is a subgroup of some right-angled Artin group and $S$ a subgroup of Thompson's group $V$ (resp. $T$, $F$). As an application, we show that several braided diagram groups embeds into Thompson's group $V$, including Higman's groups $V_{n,r}$, groups of quasi-automorphisms $QV_{n,r,p}$, and generalised Houghton's groups $H_{n,p}$., Comment: 42 pages, 15 figures. To appear in Groups Geom. Dyn

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.

In this paper we establish a Floer-theoretical analog of the classical Gysin long exact sequence from algebraic topology for circle bundles. We study algebraic and functorial properties of this sequence and derive applications to computations of Lagrangian Floer homologies as well as to questions on the topology of Lagrangian submanifolds.

For odd dimensional Poincare-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k

For a general sequence of objects and morphisms, we construct two N­ complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.

We extend the uniqueness and simplicity results of Cuntz and Krieger to the countably infinite case, under a row-finite condition on the matrix A. Then we present a new approach to calculating the K-theory of the Cuntz-Krieger algebras, using the gauge action of T, which also works when A is a countably infinite 0 − 1 matrix. This calculation uses a dual Pimsner-Voiculescu six-term exact sequence for algebras carrying an action of T. Finally, we use these new results to calculate the K-theory of the Doplicher-Roberts algebras.

The purpose of this paper is to generalize a theorem of J. F. Adams. He showed in [A] that ifX is a subcomplex of an aspherical 2-complex and the fundamental groupG ofX has no non-trivial perfect subgroups, thenX is aspherical. We weaken the hypothesis onG to “no infinite perfect subgroups”.

We relate Artin's braid groupB℞=lim→Bn to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B∞→A→F′→1 whereHkA=0,k≥1.

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l1,…,lk}⊆L such thatL is the normal closure inG of {l1,…,lk}. We prove that a [G, 2] f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e 1 2 ,…,e k 2 } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.

then it is fairly well-known that there is an exact sequence for homotopy, homology and cohomology functors which does relate T(g) and T(h) to T( f ) . We regard such an exact sequence both as a special case of the result we aim at and as the axiomatic jumping-off point for the abstract algebraic theory which is developed and applied in this paper. In a previous paper [10], to which this may be regarded as a sequel, we established the machinery of exact couples and spectral sequences in an abelian category 9~. In particular we studied the convergence problem in its fullest generality, and established the exact sequence (Theorem 4.16 of [10])

This paper is the last of a series dealing with the problem of giving an algebraic description of the torsion invariant for proper h-cobordisms using the concept of a locally finite infinite matrix. The other two papers are [l] and [6]. The reader should also consult [5]. The main results of this paper are (3.1) and (4.1) which describe the group of proper simple types on a strongly locally finite CW-complex as a Kl-type group Wh(g). The exact sequence (3.6) of [1] allows Wh(7~) to be computed in a number of cases.