19,430 research outputs found
On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups
We give an interpretation of Yetter's Invariant of manifolds in terms of
the homotopy type of the function space , where is a crossed
module and is its classifying space. From this formulation, there
follows that Yetter's invariant depends only on the homotopy type of , and
the weak homotopy type of the crossed module . We use this interpretation to
define a twisting of Yetter's Invariant by cohomology classes of crossed
modules, defined as cohomology classes of their classifying spaces, in the form
of a state sum invariant. In particular, we obtain an extension of the
Dijkgraaf-Witten Invariant of manifolds to categorical groups. The
straightforward extension to crossed complexes is also considered.Comment: 45 pages. Several improvement
Continuity properties of measurable group cohomology
A version of group cohomology for locally compact groups and Polish modules
has previously been developed using a bar resolution restricted to measurable
cochains. That theory was shown to enjoy analogs of most of the standard
algebraic properties of group cohomology, but various analytic features of
those cohomology groups were only partially understood.
This paper re-examines some of those issues. At its heart is a simple
dimension-shifting argument which enables one to `regularize' measurable
cocycles, leading to some simplifications in the description of the cohomology
groups. A range of consequences are then derived from this argument.
First, we prove that for target modules that are Fr\'echet spaces, the
cohomology groups agree with those defined using continuous cocycles, and hence
they vanish in positive degrees when the acting group is compact. Using this,
we then show that for Fr\'echet, discrete or toral modules the cohomology
groups are continuous under forming inverse limits of compact base groups, and
also under forming direct limits of discrete target modules.
Lastly, these results together enable us to establish various circumstances
under which the measurable-cochains cohomology groups coincide with others
defined using sheaves on a semi-simplicial space associated to the underlying
group, or sheaves on a classifying space for that group. We also prove in some
cases that the natural quotient topologies on the measurable-cochains
cohomology groups are Hausdorff.Comment: 52 pages. [Nov 22, 2011:] Major re-write with Calvin C. Moore as new
co-author. Results from previous version strengthened and several new results
added. [Nov 25, 2012:] Final version now available at springerlink.co
Niceness theorems
Many things in mathematics seem lamost unreasonably nice. This includes
objects, counterexamples, proofs. In this preprint I discuss many examples of
this phenomenon with emphasis on the ring of polynomials in a countably
infinite number of variables in its many incarnations such as the representing
object of the Witt vectors, the direct sum of the rings of representations of
the symmetric groups, the free lambda ring on one generator, the homology and
cohomology of the classifying space BU, ... . In addition attention is paid to
the phenomenon that solutions to universal problems (adjoint functors) tend to
pick up extra structure.Comment: 52 page
Lipshitz matchbox manifolds
A matchbox manifold is a connected, compact foliated space with totally
disconnected transversals; or in other notation, a generalized lamination. It
is said to be Lipschitz if there exists a metric on its transversals for which
the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds
include the exceptional minimal sets for -foliations of compact manifolds,
tiling spaces, the classical solenoids, and the weak solenoids of McCord and
Schori, among others. We address the question: When does a Lipschitz matchbox
manifold admit an embedding as a minimal set for a smooth dynamical system, or
more generally for as an exceptional minimal set for a -foliation of a
smooth manifold? We gives examples which do embed, and develop criteria for
showing when they do not embed, and give examples. We also discuss the
classification theory for Lipschitz weak solenoids.Comment: The paper has been significantly revised, with several proofs
correcte
Classifying community text and community groups using machine learning
The project goal is to use existing labels and supervised learning to classify groups. The labels of these groups can also be regarded as the labels of the articles in the group, because the manual labeling is also determined according to the topic of the articles in the group. Some machine learning models, such as Lightgbm anf XGBoost, are used in this project when training and predicting labels.The project goal is to use existing labels and supervised learning to classify groups. The labels of these groups can also be regarded as the labels of the articles in the group, because the manual labeling is also determined according to the topic of the articles in the group. Some machine learning models, such as Lightgbm anf XGBoost, are used in this project when training and predicting labels
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