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New Insights in Lusternik-Schnirelmann Category NIL'S 2012
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| February 17-19, 2012, Santiago de
Compostela, Spain |
Introduction Registration Schedule Abstracts Participants How to get there?
Abstracts:The following is a list of the title and abstracts provided by the speakers. Mathematics on this page is rendered by MathJax.Abstract:
Many variants of the LS category has been
given; in particular, E. Macías and H.
Colman introduced a tangential version for
foliations, where they used leafwise
contractions to transversals. In this
talk, new versions of the tangential LS
category are introduced: the measurable
category, the measurable
-category,
the topological
-category,
the secondary
-category,
the dynamical category, and the secondary
dynamical category. They will be
computed in examples, and some versions of the
classical theorems about LS category will be
shown, as well as some new theorems special of
the foliated setting.
Abstract:
Classically, an orbifold is defined as a
topological space equipped with an orbifold
structure given by an equivalence class of
orbifold atlases. From a modern point of view,
these atlases and equivalence classes are
described in terms of topological groupoids
and Morita equivalences. We show that there is
a Quillen model structure on the category of
orbifolds considered as topological groupoids,
and discuss the abstract notion of LS-category
derived from this model. This is joint work
with Cristina Costoya.
Abstract: We establish Whitehead and Ganea characterizations for proper L-S category. For that, we use a certain embedding of the proper category into an auxiliary category (exterior spaces), and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given: The characterization of proper co-H-spaces, and an analysis of the Ganea Conjecture in the proper setting. This is a joint work with P.R. García Díaz (Universidad de La Laguna) and Aniceto Murillo Mas (Universidad de Málaga). Abstract:
To resolve the systolic freedom phenomena on a
product of non-orientable manifolds, we
introduce an equivariant version of a systolic
category. In particular, we stabilize the mass
on the twisted integral homology to eliminate
the systolic freedom. For the stable mass,
there is a stable isosystolic inequality
induced from the cup-products as Gromov and
Bangert-Katz showed. This implies that the
systolic freedom from the product with a
non-orientable manifold is actually eliminated
by the stabilization process. As a corollary,
it is shown that the twisted real cup-length
gives a lower bound of the twisted stable
systolic category.
Abstract:
In this talk, I will present some
possibilities to define LS-category in the
context of abstract homotopy theory. I will
consider two different abstract frameworks,
namely J-categories in the sense of Doeraene
and monoidal cofibration categories, i.e.,
cofibration categories with a suitably
incorporated tensor product. The fundamental
properties of the LS-category can be
established in the abstract setting. Besides
the topological LS-category, the abstract
concepts cover classical algebraic
approximations of the LS-category such as
rational category, the Toomer invariant, and
the A- and M-categories of Halperin and
Lemaire.
Abstract:
We study the critical set of an arbitrary
height function in a Lie group as a tool to
compute its Lusternik-Schnirelmann category.
In particular we prove that the gradient flow
can be integrated by using the Cayley
transform. The latter also serves to obtain
explicit categorical coverings associated to
eigenvalues, thus simplifying several known
proofs. Finally we show how to extend our
results to symmetric spaces. This is a joint
work with M.J. Pereira-Sáez (University
of A Coruña).
Rational
sectional category of certain
fibrations and applications to topological
robotics
Aniceto Murillo Abstract:
The topological complexity of a
configuration space associated to a motion
planning of a given mechanical system can be
thought of as the minimum amount of
instructions of any algorithm governing the
motion. From a homotopy point of view, this is
the Lusternik-Schnirelmann sectional category
of the diagonal map on the given configuration
space. We describe in purely algebraic terms
the topological complexity of a nilpotent
rational space and present several
applications of this description. A similar
approach involving the so called reduced
topological complexity will also be presented.
Abstract:
This is an outsider's talk about recent
results obtained in collaboration with Greg
Lupton. We notice that the equality
which is well-known for topological groups
holds as well for arbitrary connected
H-spaces. It generalizes to Rudyak's higher
analogues of topological complexity. In the
second part of the talk I will try to present
a few open problems.
On the
topological complexity and the L.-S.
category of the cofibre of
the diagonal map Lucile Vandembroucq Abstract:
The topological complexity of a space has
been introduced by M. Farber in order to give
a measure of the complexity of the motion
planning problem in robotics. This invariant
is, by its definition, closely related to the
Lusternik-Schnirelmann category. By analogy to
the notion of weak (LS) category, we define
the "weak topological complexity". This
invariant is a new lower bound for the
topological complexity and turns out to be
equal to the weak category of the homotopy
cofibre of the diagonal map. As a consequence
one wonders whether the equality also holds in
the "non-weak" case. We therefore analyse the
relationship between the topological
complexity and the category of the homotopy
cofibre of the diagonal map and establish the
equality for several classes of spaces
including the H-spaces and the projective
spaces. This is joint work with J.M.
García Calcines.
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