Related papers: Proper topological complexity
We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map…
The topological classification of the inner mappings on the fully invariant regular components of the wandering set with a special attracting boundary up to the topological conjugacy is defined in terms of distinguishing graph. Two inner…
Loops are pervasive in robotics problems, appearing in mapping and localization, where one is interested in finding loop closure constraints to better approximate robot poses or other estimated quantities, as well as planning and…
The standard approach to characterizing topological matter, computing topological invariants, fails when the symmetry protecting the topological phase is preserved only on average in a disordered system. Because topological invariants rely…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are…
We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems.…
This paper explores topological complexity in the finite equivariant setting. We first define and study an equivariant version of Tanaka's combinatorial complexity for finite topological spaces. We explore the relationships between this…
We establish sharp upper bounds for the topological complexity of motion planning problem in spaces with small fundamental group, i.e. when it is finite of small order or has small cohomological dimension.
The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space…
Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…
Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
We study the integral torsion of the values of strict polynomial functors defined over the integers. We interpret some classical homological invariants as values of strict polynomial functors and therefore obtain estimates of the integral…
The main objective of this paper is to introduce a new method for qualitative analysis of various designs of robot arms. To this end we define the complexity of a map, examine its main properties and develop some methods of computation. In…
We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in…
In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for…
Topological complexity was first introduced in 2003 by Michael Farber as a homotopy invariant for a connected topological space X, denoted by TC(X). Although the invariant is defined in terms of elementary homotopy theory using well-known…