Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and f
ar between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration which mimics the smooth setting.
Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_infty colon UE to mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ om
ega colon mathrm{aut}_1(mathrm{Baut}_1(X_{mathbb Q})) to mathrm{Baut}_1(X_{mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $mathrm{Baut}_1(X_{mathbb Q})$ as a consequence. We also prove that ${mathbb C} P^n_{mathbb Q}$ cannot be realized as $mathrm{Baut}_1(X_{mathbb Q})$ for $n leq 4$ and $X$ with finite-dimensional rational homotopy groups.
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a
graph. The clique complex is a simplicial complex and its edge group is well-known to be isomorphic to the ordinary (topological) fundamental group of its geometric realization. This identification of our intrinsic digital fundamental group with a topological fundamental group---extrinsic to the digital setting---means that many familiar facts about the ordinary fundamental group may be translated into their counterparts for the digital fundamental group: The digital fundamental group of any digital circle is $mathbb{Z}$; a version of the Seifert-van Kampen Theorem holds for our digital fundamental group; every finitely presented group occurs as the (digital) fundamental group of some digital image. We also show that the (digital) fundamental group of every 2D digital image is a free group.
We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously establish
With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topology. In the ordinary topological setting, invariants such as the funda
mental group are invariants of homotopy type. In the digital setting, however, the usual notion of homotopy leads to a very rigid invariance that does not correspond well with the topological notion of homotopy invariance. In this paper, we establish fundamental results about subdivision of maps of digital images with $1$- or $2$-dimensional domains. Our results lay the groundwork for showing that the digital fundamental group is an invariant of a much less rigid equivalence relation on digital images, that is more akin to the topological notion of homotopy invariance. Our results also lay the groundwork for defining other invariants of digital images in a way that makes them invariants of this less rigid equivalence.
Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the mo
st basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.
This paper explores the relation between the structure of fibre bundles akin to those associated to a closed almost nonnegatively sectionally curved manifold and rational homotopy theory.
In arXiv:1711.10132 a new approximating invariant ${mathsf{TC}}^{mathcal{D}}$ for topological complexity was introduced called $mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${mathsf{TC}}^{mathcal{D}}$ and
the connections between ${mathsf{TC}}^{mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $mathsf{TC}$-type invariant $widetilde{mathsf{TC}}$ that can be used to give an upper bound for $mathsf{TC}$, $$mathsf{TC}(X)le {mathsf{TC}}^{mathcal{D}}(X) + leftlceil frac{2dim X -k}{k+1}rightrceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.
In this paper we study the topological invariant ${sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${sf {TC}}(X)$ is, roughly, the minimal number of contin
uous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${sf TC}(X)$ depends only on the fundamental group $pi=pi_1(X)$ and we denote it ${sf TC}(pi)$. We prove that ${sf TC}(pi)$ can be characterised as the smallest integer $k$ such that the canonical $pitimespi$-equivariant map of classifying spaces $$E(pitimespi) to E_{mathcal D}(pitimespi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{mathcal D}(pitimespi)$. The symbol $E(pitimespi)$ denotes the classifying space for free actions and $E_{mathcal D}(pitimespi)$ denotes the classifying space for actions with isotropy in a certain family $mathcal D$ of subgroups of $pitimespi$. Using this result we show how one can estimate ${sf TC}(pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${sf TC}(pi) le max{3, {rm cd}_{mathcal D}(pitimespi)},$ where ${rm cd}_{mathcal D}(pitimespi)$ denotes the cohomological dimension of $pitimespi$ with respect to the family of subgroups $mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $mathcal D$.
We prove that the sectional category of the universal fibration with fibre X, for X any space that satisfies a well-known conjecture of Halperin, equals one after rationalization.
