1,020 research outputs found
Nerve complexes of circular arcs
We show that the nerve complex of n arcs in the circle is homotopy equivalent
to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the
same even dimension. Moreover this homotopy type can be computed in time O(n
log n). For the particular case of the nerve complex of evenly-spaced arcs of
the same length, we determine the dihedral group action on homology, and we
relate the complex to a cyclic polytope with n vertices. We give three
applications of our knowledge of the homotopy types of nerve complexes of
circular arcs. First, we use the connection to cyclic polytopes to give a novel
topological proof of a known upper bound on the distance between successive
roots of a homogeneous trigonometric polynomial. Second, we show that the
Lovasz bound on the chromatic number of a circular complete graph is either
sharp or off by one. Third, we show that the Vietoris--Rips simplicial complex
of n points in the circle is homotopy equivalent to either a point, an
odd-dimensional sphere, or a wedge sum of spheres of the same even dimension,
and furthermore this homotopy type can be computed in time O(n log n)
The Vietoris-Rips complexes of a circle
Given a metric space X and a distance threshold r>0, the Vietoris-Rips
simplicial complex has as its simplices the finite subsets of X of diameter
less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian
manifold and r is sufficiently small, then the Vietoris-Rips complex is
homotopy equivalent to the original manifold. Little is known about the
behavior of Vietoris-Rips complexes for larger values of r, even though these
complexes arise naturally in applications using persistent homology. We show
that as r increases, the Vietoris-Rips complex of the circle obtains the
homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ...,
until finally it is contractible. As our main tool we introduce a directed
graph invariant, the winding fraction, which in some sense is dual to the
circular chromatic number. Using the winding fraction we classify the homotopy
types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset
of the circle, and we study the expected homotopy type of the Vietoris-Rips
complex of a uniformly random sample from the circle. Moreover, we show that as
the distance parameter increases, the ambient Cech complex of the circle also
obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the
7-sphere, ..., until finally it is contractible.Comment: Final versio
Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a
triangulated oriented marked surface S we introduce a dg-category F(S,A)
parametrizing systems of exact triangles in A labelled by triangles of S. Our
main result is that F(S,A) is independent on the choice of a triangulation of S
up to essentially unique Morita equivalence. In particular, it admits a
canonical action of the mapping class group. The proof is based on general
properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of
vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya
category of the surface S. Therefore, our construction can be seen as
implementing a 2-dimensional instance of Kontsevich's program on localizing the
Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded
version, comments welcom
Finite random coverings of one-complexes and the Euler characteristic
This article presents an algebraic topology perspective on the problem of
finding a complete coverage probability of a one dimensional domain by a
random covering, and develops techniques applicable to the problem beyond the
one dimensional case. In particular we obtain a general formula for the chance
that a collection of finitely many compact connected random sets placed on
has a union equal to . The result is derived under certain topological
assumptions on the shape of the covering sets (the covering ought to be {\em
good}, which holds if the diameter of the covering elements does not exceed a
certain size), but no a priori requirements on their distribution. An upper
bound for the coverage probability is also obtained as a consequence of the
concentration inequality. The techniques rely on a formulation of the coverage
criteria in terms of the Euler characteristic of the nerve complex associated
to the random covering.Comment: 25 pages,2 figures; final published versio
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