19 research outputs found
Inducing a map on homology from a correspondence
We study the homomorphism induced in homology by a closed correspondence
between topological spaces, using projections from the graph of the
correspondence to its domain and codomain. We provide assumptions under which
the homomorphism induced by an outer approximation of a continuous map
coincides with the homomorphism induced in homology by the map. In contrast to
more classical results we do not require that the projection to the domain have
acyclic preimages. Moreover, we show that it is possible to retrieve correct
homological information from a correspondence even if some data is missing or
perturbed. Finally, we describe an application to combinatorial maps that are
either outer approximations of continuous maps or reconstructions of such maps
from a finite set of data points
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Approximating Persistent Homology in Euclidean Space Through Collapses
The \v{C}ech complex is one of the most widely used tools in applied
algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech
filtration, the number of simplices grows exponentially in the number of input
points. A practical consequence is that computations may have to terminate at
smaller scales than what the application calls for.
In this paper we propose two methods to approximate the \v{C}ech persistence
module. Both are constructed on the level of spaces, i.e. as sequences of
simplicial complexes induced by nerves. We also show how the bottleneck
distance between such persistence modules can be understood by how tightly they
are sandwiched on the level of spaces. In turn, this implies the correctness of
our approximation methods.
Finally, we implement our methods and apply them to some example point clouds
in Euclidean space
Local cohomology and stratification
We outline an algorithm to recover the canonical (or, coarsest)
stratification of a given finite-dimensional regular CW complex into cohomology
manifolds, each of which is a union of cells. The construction proceeds by
iteratively localizing the poset of cells about a family of subposets; these
subposets are in turn determined by a collection of cosheaves which capture
variations in cohomology of cellular neighborhoods across the underlying
complex. The result is a nested sequence of categories, each containing all the
cells as its set of objects, with the property that two cells are isomorphic in
the last category if and only if they lie in the same canonical stratum. The
entire process is amenable to efficient distributed computation.Comment: Final version, published in Foundations of Computational Mathematic