21 research outputs found
Cech Closure Spaces: A Unified Framework for Discrete Homotopy
Motivated by constructions in topological data analysis and algebraic
combinatorics, we study homotopy theory on the category of Cech closure spaces
, the category whose objects are sets endowed with a Cech closure
operator and whose morphisms are the continuous maps between them. We introduce
new classes of Cech closure structures on metric spaces, graphs, and simplicial
complexes, and we show how each of these cases gives rise to an interesting
homotopy theory. In particular, we show that there exists a natural family of
Cech closure structures on metric spaces which produces a non-trivial homotopy
theory for finite metric spaces, i.e. point clouds, the spaces of interest in
topological data analysis. We then give a Cech closure structure to graphs and
simplicial complexes which may be used to construct a new combinatorial (as
opposed to topological) homotopy theory for each skeleton of those spaces. We
further show that there is a Seifert-van Kampen theorem for closure spaces, a
well-defined notion of persistent homotopy, and an associated interleaving
distance. As an illustration of the difference with the topological setting, we
calculate the fundamental group for the circle, `circular graphs', and the
wedge of circles endowed with different closure structures. Finally, we produce
a continuous map from the topological circle to `circular graphs' which, given
the appropriate closure structures, induces an isomorphism on the fundamental
groups.Comment: Incorporated referee comments, 41 page
A Topological Approach to Spectral Clustering
We propose two related unsupervised clustering algorithms which, for input,
take data assumed to be sampled from a uniform distribution supported on a
metric space , and output a clustering of the data based on the selection of
a topological model for the connected components of . Both algorithms work
by selecting a graph on the samples from a natural one-parameter family of
graphs, using a geometric criterion in the first case and an information
theoretic criterion in the second. The estimated connected components of
are identified with the kernel of the associated graph Laplacian, which allows
the algorithm to work without requiring the number of expected clusters or
other auxiliary data as input.Comment: 21 Page
Coisotropic Hofer-Zehnder capacities and non-squeezing for relative embeddings
We introduce the notion of a symplectic capacity relative to a coisotropic
submanifold of a symplectic manifold, and we construct two examples of such
capacities through modifications of the Hofer-Zehnder capacity. As a
consequence, we obtain a non-squeezing theorem for symplectic embeddings
relative to coisotropic constraints and existence results for leafwise chords
on energy surfaces.Comment: 33 pages, 4 figures; further corrections thanks to comments from the
referee; accepted for publication in Journal of Symplectic Geometr
Cofibration and Model Category Structures for Discrete and Continuous Homotopy
We show that the categories PsTop and Lim of pseudotopological spaces and
limit spaces, respectively, admit cofibration category structures, and that
PsTop admits a model category structure, giving several ways to simultaneously
study the homotopy theory of classical topological spaces, combinatorial spaces
such as graphs and matroids, and metric spaces endowed with a privileged scale,
in addition to spaces of maps between them. In the process, we give a
sufficient condition for a topological construct which contains compactly
generated Hausdorff spaces as a subcategory to admit an -category structure.
We further show that, for a topological space , the homotopy groups of
constructed in the cofibration category on PsTop are isomorphic to those
constructed classically in Top.Comment: 26 pages, corrected typos, removed some well-known results from the
preliminary Section
Kunneth Theorems for Vietoris-Rips Homology
We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a
semi-uniform space. We then interpret this result for graphs, where we show
that the Kunneth theorem holds for graphs with respect to the strong graph
product. We finish by computing the Vietoris-Rips cohomology of the torus
endowed with diferent semi-uniform structures.Comment: 10 page
Semi-coarse Spaces, Homotopy and Homology
We begin the study the algebraic topology of semi-coarse spaces, which are
generalizations of coarse spaces that enable one to endow non-trivial
`coarse-like' structures to compact metric spaces, something which is
impossible in coarse geometry. We first study homotopy in this context, and we
then construct homology groups which are invariant under semi-coarse homotopy
equivalence. We further show that any undirected graph induces a
semi-coarse structure on its set of vertices , and that the respective
semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in
turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of
undirected graphs.Comment: 53 pages. We changed 'Quasi-coarse' to 'Semi-coarse' to agree better
with conventions in the literatur
Ăclatement et contraction lagrangiens et applications
Soit (M, Ï) une variĂ©tĂ© symplectique. Nous construisons une version de lâĂ©clatement
et de la contraction symplectique, que nous définissons relative à une sous-variété
lagrangienne L â M. En outre, si M admet une involution anti-symplectique Ï, et que
nous Ă©clatons une configuration suffisament symmetrique des plongements de boules,
nous dĂ©montrons quâil existe aussi une involution anti-symplectique sur lâĂ©clatement
~M. Nous dérivons ensuite une condition homologique pour les surfaces lagrangiennes
rĂ©eles L = Fix(Ï), qui dĂ©termine quand la topologie de L change losquâon contracte une
courbe exceptionnelle C dans M. Finalement, on utilise ces constructions afin dâĂ©tudier
le packing relatif dans (âPÂČ,âPÂČ).Given a symplectic manifold (M,Ï) and a Lagrangian submanifold L, we construct
versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore,
if M admits an anti-symplectic involution Ï, i.e. a diffeomorphism such that
Ï2 = Id and Ï*Ï = âÏ , and we blow-up an appropriately symmetric configuration
of symplectic balls, then we show that there exists an antisymplectic involution on the
blow-up ~M as well. We derive a homological condition for real Lagrangian surfaces
L = Fix(Ï) which determines when the topology of L changes after a blow down, and
we then use these constructions to study the real packing numbers for real Lagrangian
submanifolds in (âPÂČ,âPÂČ)