14,395 research outputs found
On Vietoris-Rips complexes of ellipses
For a metric space and a scale parameter, the Vietoris-Rips complex
(resp. ) has as its vertex set, and a finite
subset as a simplex whenever the diameter of is
less than (resp. at most ). Though Vietoris-Rips complexes have been
studied at small choices of scale by Hausmann and Latschev, they are not
well-understood at larger scale parameters. In this paper we investigate the
homotopy types of Vietoris-Rips complexes of ellipses of small eccentricity, meaning
. Indeed, we show there are constants such that for
all , we have and , though only one of the two-spheres in is
persistent. Furthermore, we show that for any scale parameter ,
there are arbitrarily dense subsets of the ellipse such that the Vietoris-Rips
complex of the subset is not homotopy equivalent to the Vietoris-Rips complex
of the entire ellipse. As our main tool we link these homotopy types to the
structure of infinite cyclic graphs
On the homotopy type of complexes of graphs with bounded domination number
Let be the complex of graphs on vertices and domination
number at least . We prove that has the homotopy type of a
finite wedge of 2-spheres. This is done by using discrete Morse theory
techniques. Acyclicity of the needed matching is proved by introducing a
relativized form of a much used method for constructing acyclic matchings on
suitable chunks of simplices. Our approach allows us to extend our results to
the realm of infinite graphs. We give evidence supporting the assertion that
the homotopy equivalence does not generalize
as expected for , if .Comment: 19 page
On Vietoris-Rips complexes of Finite Metric Spaces with Scale
We examine the homotopy types of Vietoris-Rips complexes on certain finite
metric spaces at scale . We consider the collections of subsets of equipped with symmetric difference metric , specifically,
, ,
, and .
Here is the collection of size subsets of and
is the collection of subsets where
is a total order on the collections of subsets of and
(see the definition of in Section~\ref{Intro}). We
prove that the Vietoris-Rips complexes and
are either
contractible or homotopy equivalent to a wedge sum of 's; also, the
complexes and
are either contractible or
homotopy equivalent to a wedge sum of 's. We provide inductive formula for
these homotopy types extending the result of Barmak in \cite{Bar13} about the
independence complexes of Kneser graphs \text{KG} and the result of
Adamamszek and Adams in \cite{AA22} about Vietoris-Rips complexes of hypercube
graphs with scale
On string topology of classifying spaces
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of
the space of free loops in the classifying space of G is known to be the value
on the circle in a homological conformal field theory. This means in particular
that it admits operations parameterized by homology classes of classifying
spaces of diffeomorphism groups of surfaces. Here we present a radical
extension of this result, giving a new construction in which diffeomorphisms
are replaced with homotopy equivalences, and surfaces with boundary are
replaced with arbitrary spaces homotopy equivalent to finite graphs. The result
is a novel kind of field theory which is related to both the diffeomorphism
groups of surfaces and the automorphism groups of free groups with boundaries.
Our work shows that the algebraic structures in string topology of classifying
spaces can be brought into line with, and in fact far exceed, those available
in string topology of manifolds. For simplicity, we restrict to the
characteristic 2 case. The generalization to arbitrary characteristic will be
addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic
Hom complexes and homotopy theory in the category of graphs
We investigate a notion of -homotopy of graph maps that is based on
the internal hom associated to the categorical product in the category of
graphs. It is shown that graph -homotopy is characterized by the
topological properties of the \Hom complex, a functorial way to assign a
poset (and hence topological space) to a pair of graphs; \Hom complexes were
introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give
topological bounds on chromatic number. Along the way, we also establish some
structural properties of \Hom complexes involving products and exponentials
of graphs, as well as a symmetry result which can be used to reprove a theorem
of Kozlov involving foldings of graphs. Graph -homotopy naturally leads
to a notion of homotopy equivalence which we show has several equivalent
characterizations. We apply the notions of -homotopy equivalence to the
class of dismantlable graphs to get a list of conditions that again
characterize these. We end with a discussion of graph homotopies arising from
other internal homs, including the construction of `-theory' associated to
the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J.
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