1,720,964 research outputs found
Amenability and acyclicity in bounded cohomology
Full text linkJohnson's characterization of amenable groups states that a discrete group is amenable
if and only if for all dual normed -modules .
In this paper, we extend the previous result to homomorphisms by proving the converse of the
\emph{Mapping Theorem}:
a surjective group homomorphism has amenable kernel if and only if the induced inflation map
is an isometric isomorphism for every dual normed -module
.
In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms
with the property that the inflation maps in bounded cohomology are isometric isomorphisms for \emph{all} Banach -modules.
Finally, we also prove a characterization of the (larger) class of \emph{boundedly acyclic} homomorphisms, that is, the class of group homomorphisms for which the restriction maps in bounded cohomology are isomorphisms for a suitable family of dual normed -modules including the trivial -module . We then extend the first and third results to topological spaces and obtain characterizations of \emph{amenable} maps and
\emph{boundedly acyclic} maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices
of coefficients
Gromov’s theory of multicomplexes with applications to bounded cohomology and simplicial volume
Full text linkThe simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets.
In the first part of this paper we lay the foundation of the theory of multicomplexes. We construct the singular multicomplex K(X) associated to a topological space X, and we prove that K(X) is homotopy equivalent to X for every CW complex X. Following Gromov, we introduce the notion of completeness, which translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes.
In the second part we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. We provide complete proofs of Gromov's Mapping Theorem (which implies that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The third part is devoted to the study of locally finite chains on non-compact spaces. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the l^1-invisibility of closed manifolds in terms of amenable covers. As an application, we give the first complete proof of the vanishing of the simplicial volume of the product of three open manifolds
Amenable category and complexity
Full text linkAmenable category is a variant of the Lusternik-Schnirelman category, based
on covers by amenable open subsets. We study the monotonicity problem for
degree-one maps and amenable category and the relation between amenable
category and topological complexity.Comment: 33 pages; to appear in AG
Bounded cohomology and binate groups
Full text linkA group is boundedly acyclic if its bounded cohomology with trivial real
coefficients vanishes in all positive degrees. Amenable groups are boundedly
acyclic, while the first non-amenable examples were the group of compactly
supported homeomorphisms of (Matsumoto--Morita) and mitotic
groups (L\"oh). We prove that binate (alias pseudo-mitotic) groups are
boundedly acyclic, which provides a unifying approach to the aforementioned
results. Moreover, we show that binate groups are universally boundedly
acyclic.
We obtain several new examples of boundedly acyclic groups as well as
computations of the bounded cohomology of certain groups acting on the circle.
In particular, we discuss how these results suggest that the bounded cohomology
of the Thompson groups , , and is as simple as possible.Comment: 33 pages, one figure; v3: refs updated and minor changes. The new
paragraph 3.1.4 contains examples of amenable binate groups. To appear in J.
Aust. Math. So
On the simplicial volume and the Euler characteristic of (aspherical) manifolds
Full text linkA well-known question by Gromov asks whether the vanishing of the
simplicial volume of oriented closed aspherical manifolds
implies the vanishing of the Euler characteristic.
We study various versions of Gromov's question and collect
strategies towards affirmative answers and strategies towards
negative answers to this problem. Moreover, we put Gromov's question
into context with other open problems in low- and high-dimensional
topology.
A special emphasis is put on a comparative analysis of the
additivity properties of the simplicial volume and the Euler
characteristic for manifolds with boundary. We explain that the
simplicial volume defines a symmetric monoidal functor (TQFT) on the emph{amenable} cobordism category, but not on the
whole cobordism category. In addition, using known computations of
simplicial volumes, we conclude that the fundamental group of the
4-dimensional amenable cobordism category is not finitely
generated.
We also consider new variations of Gromov's question. Specifically,
we show that counterexamples exist among aspherical spaces that are
only homology equivalent to oriented closed connected manifolds
Amenable covers and integral foliated simplicial volume
Full text linkIn analogy with ordinary simplicial volume, we show that integral foliated
simplicial volume of oriented closed connected aspherical -manifolds that
admit an open amenable cover of multiplicity at most is zero. This implies
that the fundamental groups of such manifolds have fixed price and are cheap as
well as reproves some statements about homology growth.Comment: 23 pages, revised version according with the referee's comments. To
appear in New York J. Mat
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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